# Artificial Intelligence across Europe: A Study on Awareness, Attitude and
Trust

Teresa Scantamburlo, Atia Cortés, Francesca Foffano, Cristian Barrué,
Veronica Distefano, Long Pham, Alessandro Fabris

(15 May 2023)

###### Abstract

This paper presents the results of an extensive study investigating the
opinions on Artificial Intelligence (AI) of a sample of 4,006 European
citizens from eight distinct countries (France, Germany, Italy, Netherlands,
Poland, Romania, Spain, and Sweden). The aim of the study is to gain a better
understanding of people’s views and perceptions within the European context,
which is already marked by important policy actions and regulatory processes.
To survey the perceptions of the citizens of Europe we design and validate a
new questionnaire (PAICE) structured around three dimensions: people’s
awareness, attitude, and trust. We observe that while awareness is
characterized by a low level of self-assessed competency, the attitude toward
AI is very positive for more than half of the population. Reflecting upon the
collected results, we highlight implicit contradictions and identify trends
that may interfere with the creation of an ecosystem of trust and the
development of inclusive AI policies. The introduction of rules that ensure
legal and ethical standards, along with the activity of high-level educational
entities, and the promotion of AI literacy are identified as key factors in
supporting a trustworthy AI ecosystem. We make some recommendations for AI
governance focused on the European context and conclude with suggestions for
future work.

##  1 Introduction

In April 2021 the European Commission (EC) proposed a set of rules to regulate
Artificial Intelligence (AI) systems operating across Europe [22]. This was an
important step in a long-term process in which the European Union developed
its approach towards AI [48], setting up policy agendas [18] and ethics
guidelines [35] among others. During this period, the EC sought feedback from
different stakeholders to ensure an inclusive policy development, such as the
consultation run from February to June 2020 to gather opinions about the White
Paper on AI [20]. Usually, these consultations solicit reflection on specific
actions or policy proposals and can reveal partial information, if anything,
about what people think about AI and its related impact on society.

Knowing people’s views and perceptions is key to deploying effective
governance mechanisms and integrating rules into society. In this paper, we
aim to fill this gap and report the results of a survey investigating the
knowledge and perception of AI in Europe. For this reason, we designed,
developed, and validated a new questionnaire, the Perceptions on AI by the
Citizens of Europe questionnaire (PAICE), structured around three dimensions:
Awareness, attitude, and trust. Based on a computer-assisted web interview
methodology (CAWI) we collected and analyzed the opinions of 4,006 European
citizens from eight countries (France, Germany, Italy, Netherlands, Poland,
Romania, Spain, and Sweden), stratified by age, gender, and geographic
regions.

The collected responses show that respondents’ self-assessed knowledge about
AI is low, while their attitude is very positive and slightly varies depending
on the context of use (e.g. approval is lower when AI is applied to human
resources management). The most important measures to increase trust in the AI
ecosystem include the introduction of laws by national authorities,
transparent communication by AI providers, and education activities. Among
trusted entities that could ensure a beneficial use of AI, universities and
research centers are ranked higher than other organizations (e.g. national
governments and tech companies). The statistical analysis shows that the
questionnaire has good internal consistency and that the validity is adequate.

We analyze the results of the survey and identify a few contrasting
perceptions which may reflect three broader social trends: 1) approval of a
hyped, but poorly known, technology; 2) disconnect from public AI policies; 3)
poor engagement with AI education and training. We discuss how these trends
may create friction in the creation of a Trustworthy AI culture and suggest a
few recommendations. Our findings call for greater consideration of people’s
views and participation in AI policy-making, especially if we consider the
rapid transformations introduced by AI into society and the abundance of
policy efforts by states and intergovernmental organizations [57, 22, 10, 65,
31, 62].

###  1.1 Related work

AI and trust recall a vast academic literature investigating shared principles
among ethics guidelines [33, 29, 36], as well as challenges and future
directions [61, 52, 37, 30, 49]. In this section, we focus on previous surveys
analyzing citizen awareness, trust, and attitude towards AI from different
perspectives.

In a global study surveying 10,000 citizens spanning eight countries across
six continents [43], respondents reported a mix of positive and negative
feelings about AI. In a similar study, the UK expressed a markedly negative
view of AI, while showing a reasonable understanding and awareness of this
technology [8]. The US population has been surveyed on a key dimension of
trust: the perception of governance [66, 67]. While most people (especially
older segments) find the issue very important, they state that they have
little trust in the actors who have the power to develop and manage AI (e.g.
companies, universities, US agencies). Another US-related work investigated
the ethical preferences of different groups of people and found that AI
practitioners’ value priorities differ from those of the general public [38].

Studies focused on the perception of AI in Europe are not entirely new. In an
EU-wide survey, the authors focused on a notion of AI centered around
robotics, finding attitudes to be generally positive, with concerns related to
job losses [17], later confirmed in a follow-up study [21]. These are generic
studies of EU public opinion about science and technology, with only a
marginal focus on AI. A subsequent survey on opinion about AI highlighted
discrimination and lack of accountability as key concerns for European
citizens, and a belief that public policy intervention is needed, shared by a
majority of respondents [19].

Recently, [44] analyzed the positive and negative expectations of 164
individuals visiting a Science Gallery exhibition in Dublin. The study found
that awareness of AI is relatively good, opportunities are related to economic
growth and social progress (e.g. mentioning the positive impact on medicine,
science, and environments) and concerns are connected to automation, followed
by privacy and surveillance. [60] examined awareness of AI, emotional
responses to narratives, and the perceived likelihood of future scenarios in
Italy. The authors pointed out a positive correlation between the level of
digital expertise and general knowledge of AI and showed an important gender
divide with respect to the emotional response to narratives with women more
concerned than men across all scenarios. [46] investigated how German people
prioritize different ethical principles (transparency, fairness, non-
maleficence, responsibility, beneficence, privacy, and machine autonomy) with
regard to the application of AI to fraud detection. The study found that all
ethical principles are equally important for the respondents but different
preference profiles for ethically designed systems exist.

###  1.2 Research questions

The present work departs from the existing literature in two fundamental ways.
First, it takes AI as its main target not as part of broader investigations in
science and technology [21], connecting different perspectives (such as
awareness and trust) with specific use cases. Second, it aims at reaching a
large population involving more than one European country or demographics [46,
60, 44].

The questionnaire was developed in the context of an Horizon 2020 project by a
multidisciplinary team of researchers. The research questions addressed by the
team are the following:

  * •

RQ1: _To what extent are EU citizens familiar with AI and the surrounding
debate?_ This covers aspects concerning citizens’ awareness and competency
such as: what people think they know about AI, where they think AI is applied,
what is the perceived impact of AI, and which EU initiatives addressing
ethical and legal concerns they are aware of.

  * •

RQ2: _To what extent do EU citizens approve AI?_ This research question
connects to citizens’ attitude towards AI and its use in some specific sectors
or contexts of application (such as job recruitment).

  * •

RQ3: _What could contribute to increasing citizens’ trust?_ This question
investigates citizens’ priorities to promote the responsible development of AI
in terms of actions, actors and ethical requirements.

These questions guided the development of the questionnaire around the
dimensions of awareness, attitude, and trust. The structure of the
questionnaire was also explored in our analysis (i.e. validity and
reliability). This allowed us to identify which items of the questionnaire can
be used to validate the dimensions suggested by the team of experts who
designed the research instrument.

The rest of the paper is organized as follows. First, we present the
methodology guiding the survey design. Next, we report the results of the
survey according to the dimensions of interest (i.e. awareness, attitude, and
trust), and analyze the validity and reliability of the questionnaire. We
discuss the results pointing out implicit tensions and discussing potential
barriers to the development of inclusive AI policy processes, thereby making
recommendations to improve current efforts, especially at the European level.
Finally, we summarise our findings and suggest future research directions.

## Methods and materials

### Survey method

This survey was conducted by the market research agency Marketing Problem
Solving (MPS) based in Italy [55]. The survey was carried out through online
interviews (CAWI) on the basis of a structured questionnaire. The average
completion time was 20 minutes. MPS programmed the script of the questionnaire
through the creation of a website hosted on the web server owned by MPS and
managed the data collection process.

The invitation to fill out the questionnaire was sent by e-mail to members of
an online panel who voluntarily agreed to share their opinions. To facilitate
the task, panel members received the questionnaire in their own language. The
respondents were free to drop out at any point and had the opportunity to go
back to previous items and change their answers. Respondents’ information was
recorded in compliance with the General Data Protection Regulation (GDPR) and
the Italian legislation on data protection and privacy.

From the 1st to the 15th of June 2021 MPS, realized a total of 4,006
interviews in eight European countries: France, Germany, Italy, Netherlands,
Poland, Romania, Spain, and Sweden. Countries were selected with a view to
cover different European regions (southern, central/eastern, northern, and
western). The survey was completed by individuals aged between 18 and 75
years. Quotas were imposed to ensure the representativeness of the sample with
respect to gender, age group (18-34, 35-54, 55-75), and geographical area of
residence.

Before undertaking the survey, MPS tested the questionnaire with a sample of
panel members to assess the clarity of instructions and the average completion
time. MPS monitored the whole interview process to ensure the quality of
responses, e.g. by removing participants who completed the survey too quickly
or provided contradictory answers.

The original version of the survey was developed and revised in English and
then translations in other languages (Italian, Spanish, German, Polish,
French, Romanian, Dutch, Swedish) were made by professional translators or
mother tongue experts.

Table 1: Description of the survey population Country | Gender | Age groups | City size by population
---|---|---|---
woman | man | 18-34 | 35-54 | 55-75 |  $<$ 10K | 10K-100K |  $>$ 100K
$France$ | 48.5% | 51.5% | 28.9% | 38.4% | 32.7% | 42.2% | 33.9% | 23.9%
$Germany$ | 50.1% | 49.9% | 27.9% | 38.3% | 33.7% | 30.2% | 36.7% | 33.1%
$Italy$ | 48.2% | 51.8% | 23.8% | 41.0% | 35.2% | 22.8% | 51% | 26.2%
$Netherlands$ | 50.2% | 49.8% | 28.6% | 36.6% | 34.8% | 19% | 49% | 32%
$Poland$ | 49.0% | 51.0% | 30.8% | 37.4% | 31.8% | 24.2% | 31.6% | 44.2%
$Romania$ | 50.0% | 50.0% | 27.5% | 40.9% | 31.5% | 29.3% | 28.2% | 42.5%
$Spain$ | 49.7% | 50.3% | 21.6% | 44.5% | 33.9% | 19% | 41.1% | 39.9%
$Sweden$ | 50.0% | 50.0% | 31.9% | 37.9% | 30.1% | 12.4% | 45.5% | 42.1%

###  1.3 Population

To obtain a random stratified sample, members of the population were first
divided into non-overlapping subgroups of units called _strata_ (country),
then, a sample was selected from each stratum based on geographic regions
(unit of analysis), age groups and gender through a simple random sampling.
The sample was made up of 4,006 individuals with equal representation for each
country (12.5%).

The sample was composed of individuals in the age range 18-75 (mean age = 45,
std = $\pm{14.83}$) where women were $49.3\%$ (mean age = 46, std =
$\pm{14.92}$) and men $50.4\%$ (mean age = 45.5, std = $\pm{14.83}$). Note
that in our analysis we considered only male and female groups since the
respondents choosing the option “others” were only 0.3% of the whole
population. The age groups were coded into three levels: young (18-34 years),
middle age (35-54), and senior (55-75) people. In particular, 26.5% of
respondents were young, 39.4% were middle age, and the remaining 33% were
senior. We also investigated the population size of the place of residence and
found that 25% of respondents lived in a city with a population up to 10K,
about 40% lived in a city with 10-100K inhabitants, and the remaining 35%
lived in a large city with more than 100K inhabitants. Information about
gender, age groups, and city size related to respondents of each country is
summarized in Table 1.

With reference to formal education, the descriptive analysis highlighted that
40% of the respondents had the highest level of formal education (bachelor,
master, or doctoral degree). Note that this percentage is higher than the
share of European citizens with tertiary education (i.e. also including trade
schools and vocational education) which is estimated at 31% [23]. The choice
of the survey methodology, based on online interviews, possibly facilitated
the participation of subjects with higher levels of education.

To investigate confidence with Information and Communication Technology
domains we submitted to the respondents an item assessing their level of
competence in digital skills on a five-point ordinal scale from almost no
knowledge to advanced knowledge. It was observed that 44% of the respondents
have an intermediate level of competence in digital skills. Among those who
feel less competent in digital skills, we found French and German respondents
who represent respectively 31.7% and 34.5% of the population surveyed in each
country. The countries reporting the highest level of competency are Spain and
Italy where respondents with intermediate or advanced knowledge are 82.9% and
79.8% respectively. For more details on digital skills and formal education
see tables “digital skills” and “education” in the supplemental material
(Digital skills).

###  1.4 Questionnaire design

The PAICE questionnaire was created by a group of researchers from different
backgrounds (AI & Computer Science, Philosophy, Engineering, Psychology, and
Communication) including the authors of the present work.

The design of the questionnaire took six months, from January to June 2021,
during which the group met on a monthly basis. In the early stages of the
design process, the group collected and analyzed the existing literature and
previous surveys at a European and worldwide level. Based on the literature
review, the group identified the research questions and subsequently defined
the questions for the research instrument. After a refinement process, the
group agreed on a total of 14 items including Likert scale, dichotomous,
multi-response items, and ranking. The items were organized according to the
three dimensions introduced above (awareness, attitude, and trust) with a view
to address the starting research questions. An overview of the structure of
the questionnaire with question types and the topics of each item is reported
in Table 2. Note that some questions, since they applied to different sectors,
policy measures, or entities, were split into sub-items (e.g. Q7_1 to Q7_10).

In addition, the questionnaire presented: a control question about the
perceived impact of AI, a question investigating the interest in attending a
free course on AI, and seven questions on socio-demographic aspects (i.e. age
group membership, gender, geographical area, population size, job sector,
level of education, and digital expertise). The control question, which was a
repetition of item Q3 (see Table 2), was added to assess possible changes in
opinions after the completion of the questionnaire. The English version of the
full questionnaire is available in the supplemental material (Questionnaire).

Likert scale items ranged from 1 to 5 where 1 referred to negative or low
values (e.g. “not at all”, “never”, “not important at all” and “strongly
disapprove”) and 5 to positive or high values (e.g. “a lot”, “always”, “very
important” and “strongly approve”). For item Q5 we also added the option “I
don’t know” to accommodate respondents who did not have a clear opinion on the
topic (awareness of interaction). We chose the 5-point Likert scale because
this is largely used in social science research to study human attitudes and
perceptions [56]. Though the optimum number of choices in a Likert-type scale
is a subject of dispute [41], we opted for a 5-point scale to ensure items’
simplicity and intelligibility [51, 3].

To offer a common ground to all respondents we introduced the following
definition of Artificial Intelligence at the beginning of the questionnaire:
“Artificial intelligence (AI) refers to computer systems that can perform
tasks that usually require intelligence (e.g. making decisions, achieving
goals, planning, learning, reasoning, etc.). AI systems can perform these
tasks based on objectives set by humans with a few explicit instructions.”
Given the heterogeneity of the consulted population, we chose a simple
definition that could be intelligible by a large audience.

Table 2: PAICE questionnaire design and structure | Question Type | Description
---|---|---
Awareness |  Likert scale |  Q1: Knowledge about AI Q3: Impact of AI on daily life (repeated for control question) Q5: Awareness of interaction with products incorporating AI Q7: Awareness of the application of AI in different sectors across Europe
Dichotomous |  Q4: Knowledge about three specific European initiatives: the General Data Protection Regulation (GDPR), the Ethics Guidelines for Trustworthy AI, the proposal of an AI regulation
Multi-response |  Q6: Awareness of products embedding AI
Attitude |  Likert scale |  Q2: General attitude towards AI Q8: Attitude towards the application of AI in specific sectors Q9: Perceived comfort with a scenario applying AI to job recruitment Q10: Perceived comfort with a scenario applying AI to energy consumption
Trust |  Likert scale |  Q12: Importance of specific policy measures to increase trust Q13: Importance of education to increase trust in AI Q14: Trust in entities that may ensure a beneficial use of AI
Ranking |  Q11: The three most important ethical requirements derived from [35] in relation to the aforementioned scenarios (i.e. Q9 and Q10)

###  1.5 Statistics

To explore the theoretical dimensions structuring the PAICE (awareness,
attitude, and trust) an Exploratory Factor Analysis (EFA) and confirmatory
factor analysis (CFA) were performed. The aim was to evaluate the robustness
of items in the questionnaire. To do this we randomly split the sample
(n=4,006) into two groups n=2,450 for EFA and n=1,051 for CFA. Note that only
items measured on the Likert scale were included in this analysis.

The EFA was performed to determine the number of fundamental (latent)
constructs underlying the set of items and quantify the extent to which each
item is associated with the construct [64]. In this context, the EFA allows us
to study the strength of relations between the dimensions identified by the
team of experts who designed the questionnaire and the associated items.
Before performing the EFA analysis, two criteria were tested to determine
whether factor analysis was appropriate: the Kaiser-Meyer-Olkin (KMO) measure
of sampling adequacy and Barlett’s test of sphericity assumptions [42, 24]. A
KMO index $>0.7$ and a Barlett’s test of sphericity p-value $<0.05$ are
considered appropriate values to conduct the EFA [42, 24]. The implementation
of the EFA was based on a polychoric correlation matrix since the
questionnaire is composed of ordinal items (i.e. Likert scale). The EFA was
run by using principal axis factoring, because it does not assume normality of
data [7], with oblique rotation. The parallel analysis was used to identify
the optimum number of factors to be retained. We also assessed inter-factors
correlation, in order to evaluate if some theoretical dimensions are
correlated strongly with each other, i.e. $>0.7$.

To assess the internal consistency of the EFA solution, we calculated
Cronbach’s $\alpha$ and ordinal $\alpha$ which is considered the most
appropriate coefficient for ordinal-type scales [28, 68]. These indices take
values in the range [0, 1], so the internal consistency is acceptable if the
indices are greater than 0.80 [56].

Finally, the validity of the factor structure derived from the EFA was
evaluated by using the CFA. The implementation for the CFA was based on a
polychoric matrix and the robust diagonally weighted least squares (RDWLS)
extraction method which is more suitable for ordinal data than other
extraction methods [50].

We assessed the fit of the model using the following criteria: root mean
squared error of approximation (RMSEA$<0.08$), the comparative fit index (CFI)
and the Tucker-Lewis index (TLI) with values above 0.95 and 0.90,
respectively, and the standardized root mean residual (SRMR$<0.08$) [63]. All
analyses were performed by using the statistical package for social science
V.25.0.2 (SPSS) and R version 3.4.1 (The R Foundation for Statistical
Computing, Vienna, Austria).

### Limitations

This work, like any other, has some inherent limitations. Though we tried to
represent different European regions, the sample does not cover all European
countries. Thus, our analysis may not be representative of the opinions of all
EU citizens. As we suggest in the conclusion, extending the questionnaire to
other countries will give a more complete picture of European society.
Moreover, our questionnaire administration methodology (CAWI) assumes that the
target population has access to the internet and is familiar with web
navigation. This choice could have indirectly impacted on the selection of the
population interviewed. The latter may be skewed towards people with higher
education levels and/or wealthier socio-economic status. Another limitation
concerns the measurement of awareness. In this study, we focus on self-
reported awareness, which may suffer from subjective and contextual factors.
Objective knowledge about AI is another important dimension of awareness; its
rigorous measurement would require the development of a specific methodology
going beyond the scope of this work.

##  2 Results

The responses to the questionnaire are presented with respect to the three
dimensions: Awareness, attitude, and trust. Aggregated responses to all items
are reported in tables “Likert-scale items” and “Non-Likert scales” with the
descriptive statistics in the supplemental material (Responses). Responses
were compared with respect to different groups by using Kruskal-Wallis test
where a p-value $<0.01$ is considered statistically significant. In our
comparison, we considered the following groupings: Countries, age groups, and
gender. With respect to countries, we found statistically significant
differences among groups for all Likert scale items. As for age and gender
groups, we found statistically significant differences for a subset of items.
For the sake of brevity, in the subsequent sections, we will comment only most
significant differences. The results of statistical tests are reported in
tables “Awareness”, “Attitude”, and “Trust” in the supplemental material
(Statistics by Groups).

Finally, we report the results of the analysis performed to assess the
questionnaire’s validity and reliability.

###  2.1 Awareness

Figure 1: Responses to Likert scale items associated with awareness. Low-scale
values (1 and 2) are represented by red-like colors, while high-scale values
(4 and 5) are represented by blue-like colors. Item Q7 is split into sub-items
regarding the perceived presence of AI in ten different sectors.

In Figure 1, we represented the percentages of responses to Likert scale items
connected to awareness. Blue and red colored segments identify the two extreme
positions: high and low levels of awareness respectively. The largest red
segment, including the lowest scale values (i.e. 1 and 2), regards the self-
assessed competency on AI (Q1). In this item, almost half of the respondents
(49.5%) reported having low or no knowledge, while only 20.9% considered their
knowledge to be advanced or expert-level.Analyzing the results by country,
Germany and the Netherlands have the highest percentage of respondents who
feel less knowledgeable, at 66% and 63% respectively. If we look at gender,
the percentage of individuals who feel less competent is greater for males
(55%) as compared to females (43%). With respect to age, the portion of
individuals with low or no competency is higher for seniors (63%) and lower
for young respondents (32%).

When asked about being aware of interacting with a product or service based on
AI (Q5), only 26.5% reported being often or always aware, while 24.7% reported
to be never or seldom aware, and 12.6% chose the “I don’t know” option. In
Germany the fraction of people who feel never or seldom aware increases by 9
percentage points (32%). Male respondents declared a higher rate of low or no
awareness of interaction (25%) as compared to females (23%). The group of
senior respondents achieved the highest percentage of answers expressing
unawareness during interaction (28%).

In relation to the impact of AI in their daily lives (Q3), half of the
respondents (53.4%) felt like it has somewhat or a lot of impact,while 16.7%
answered with “not so much” or “not at all”. The perception of (high) impact
is greater in Spain (73%) and lower in Poland (33%) - the latter is also the
country in which there is the highest fraction of answers reporting a low
perceived impact of AI on their lives (29%).

Items from Q7_1 to Q7_10 assessed to what extent respondents feel AI is used
in distinct sectors across Europe. Military (67.9%) and Manufacturing (66.5%)
present a higher fraction of respondents perceiving AI as being somewhat or
very present in such sectors. On the other hand, Human Resources (50.1%) and
Agriculture (51.4%) present a lower perception of the presence of AI.

Regarding respondents’ familiarity with the normative and ethical European
framework (Q4), two out of three respondents (65.6%) have heard about GDPR,
while only one out of three were aware of the Trustworthy AI Guidelines or the
AI Act (28.3% and 29.8% respectively).

Participants were also introduced to a list of applications and were asked
about which ones may contain AI components (Q6). Facial recognition apps,
content and product recommendations, search engines, traffic navigation apps,
and car ride-sharing apps were the most identified applications, selected by
half of the respondents. Other options with more limited AI applications, such
as calculators or text editors, were included by 32.6% and 26.3% of
participants respectively. Finally, 7.2% of respondents selected the option
“none of the above”, hence did not identify any AI-based application.

###  2.2 Attitude

Figure 2: Responses to Likert scale items associated with attitude. Low-scale
values (1 and 2) are represented by red colors, while high-scale values (4 and
5) are represented by blue colors. Item Q8 is split in sub-items regarding the
attitude towards AI in ten different sectors.

In Figure 2, we reported the percentages of responses to Likert scale items
associated with attitude, where blue segments represent a (very) positive
inclination and red segments indicate a (very) negative one.

Regarding their general attitude towards AI (Q2), 63.4% of the respondents
report strongly approving or approving of AI. The most receptive countries
were Romania and Spain with almost 80% approval, while in France fewer than
50% participants declared approval of AI. With respect to gender, females
expressed to be more positive as compared to males, with approval or strong
approval at 68% and 59% respectively. When considering age, the class of
younger respondents reached the highest rate of approval (70%), while the
group of seniors reported the lowest one (58%).

Items from Q8_1 to Q8_10 aimed to further understand how approval varies
depending on the sector of application. Law Enforcement and Environment have
the highest acceptance with an average of 67% of participants opting for
approval or strong approval, followed by Manufacturing, Healthcare, and
Agriculture. Human resources presents the lowest acceptance rate (47.3%) and
the highest dissatisfaction rate with 21.2% of respondents disapproving or
strongly disapproving.

We also considered two specific use case scenarios: Q9 presents an AI-based
system that screens candidates’ resumes and selects those who can access the
interviewing stage [4]; Q10 introduces a smart meter to reduce energy
consumption inspired by _demand side management_ [45] that leverages AI to
recommend more efficient usage and provide personalized offers from energy
providers. While the proportion of neutral positions is approximately the same
for both scenarios, the approval is significantly higher for the smart meter
with 58.3% of the respondents feeling fairly or very comfortable, as opposed
to 44.7% for the resume screening system. Again, we observed statically
significant differences among countries. Poland is the most receptive country
with about 67% of respondents feeling fairly or very comfortable in both
scenarios. The trend for gender and age groups is similar to that found for
general attitude with a preference for the smart meter scenario.

###  2.3 Trust

In Figure 3, we represented the responses to Likert scale items referred to
trust. Similarly to previous dimensions, colors are indicative of respondents’
satisfaction with actions and entities aimed to ensure trust. When asked to
assess the importance of a set of policy measures to increase trust (Q12), 76%
of the respondents valued as important or very important the deployment of a
set of laws by a national authority that guarantees ethical standards and
social responsibility in the AI application. Romania and Germany are the
countries in which this percentage is the largest, at 90% and 82%
respectively. With regard to age, a large proportion of senior respondents
consider this measure important (81%), followed by young (71%) and the middle
age respondents (68%). The remaining measures were also highly supported (more
than 50%); the least valued one was the creation of diverse design teams and
the consultation of different stakeholders throughout the entire life cycle of
the AI product (64.4%). Education as a remedy to improve citizens’ trust (Q13)
was also largely approved with 71.4% of agreement or strong agreement. Note
that this percentage increases significantly in Romania and Spain where
agreement reaches 85% and 83% respectively, while it falls to 59% in France.

With respect to trusted entities ensuring a beneficial use of AI (Q14), two
out of three participants (67%) rated universities and research centers as
entities that could be trusted a lot or somewhat. Note that this percentage
varies across countries with Romania reporting the highest value (77%) and
France the lowest one (55%). Social media companies are the least trusted
entity with only 35% of respondents trusting them. With respect to countries
the percentage is higher in Italy (47%) and lower in the Netherlands (26%),
while, if we consider age groups, trust in social media is lowest for senior
respondents (24%) and highest for young respondents (46%).

With Q11, we asked to select three out of the seven most important aspects
that an organization should consider to developing or using AI in relation to
the previous scenarios (Q9 and Q10). Interestingly, there is a clear
preference towards technical aspects related to security, robustness and human
oversight, with Privacy and Data Protection leading as a choice for 30.8%
respondents. On the other hand, the Societal and Environmental impact of AI
applications was only selected by the 5% of the respondents as a first or
second choice.

Figure 3: Responses to Likert scale items associated to trust. Low-scale
values (1 and 2) are represented by red-like colors, while high-scale values
(4 and 5) are represented by blue-like colors. Item Q12 is split into sub-
items regarding the perceived importance of six different policy measures.
Item Q14 is split in sub-items related to the perceived trust in six different
entities.

###  2.4 Questionnaire validity and reliability

Among the 4,006 participants, 501 ticked the response option “I don’t know”
for item Q5 (i.e. “How often are you aware of interacting with a
product/service based on or including AI?”), corresponding to 12.6% of the
sample. Therefore, these responses were excluded from the statistical
analysis. A qualitative analysis was conducted to explore the content of each
item and identify the ones with multicollinearity issues [12, 40].

The Kaiser–Meyer–Olkin (KMO) test and the Bartlett’s test of sphericity showed
that the data are appropriate to perform the EFA with a KMO index $=0.93$ and
a Barlett’s test of sphericity p-value $<0.0001$ [42, 24]. Parallel analysis
suggested three factors which are detailed in the supplemental material
(Exploratory Factor Analysis: Figure). The three factors accounted for 62% of
the total variance. In particular, we extracted the factors based on factor
loading and the interpretability of the factors. Note that, the items with low
factor loading ($<0.50$) were not considered while the remaining items were
assigned to a single factor according to their highest loading (see
supplemental material (Exploratory Factor Analysis: Table).

The items that load on the same factor suggested that factor 1 (26% of the
total variance) refers to awareness and includes 7 items (Q7_2, Q7_4, Q7_5,
Q7_6, Q7_7, Q7_9, Q7_10); factor 2 (25% of the total variance) refers to
attitude and includes 6 items (Q8_1, Q8_2, Q8_3, Q8_4, Q8_6, Q8_7) and factor
3 (10% of the total variance) refers to trust and includes only 3 items
(Q14_2, Q14_4, Q14_6).

We also assessed the factor correlation matrix of the final EFA to assess the
discriminant validity. The correlations between all three factors were found
positive. The largest positive correlation was between factor 1 and factor 2
(0.52), and the smallest correlation was between factor 2 and factor 3 (0.37).
Hence, we did not find correlation coefficients greater than 0.7; therefore
the factors derived from EFA revealed adequate discriminant validity among the
factors.

For reliability, both Cronbach’s $\alpha$ and ordinal $\alpha$ were found to
be large enough ($\alpha>0.8$), indicating that the questionnaire had good
internal consistency. Then, we used the CFA to examine the proposed factorial
structure of the PAICE. Overall, our CFA results showed that the EFA model
showed acceptable fit indices (RMSEA (90%CI = 0.011; CFI = 0.99; TLI = .99,
SRMR = 0.03; p-value $<0.001$).

##  3 Discussion

The collected responses reveal some contrasts that are worthy of in-depth
analysis. These tensions may signal friction in current efforts towards a
Trustworthy AI innovation and, in particular, call for reflection on the EU
context, where the AI strategy aims to build an ecosystem of trust and the
development of an AI regulation is underway. Note that these contrasts reflect
more implicit contradictions rather than disagreements openly expressed. Yet,
pointing them out allows us to discuss critical social orientations that may
constitute a barrier to the development of a trustworthy AI culture and, most
importantly, an inclusive approach to AI governance.

###  3.1 Implicit contradictions

#####  Knowledge about AI _vs_ Approval of AI.

The first remarkable result of this survey is that respondents’ (self-
assessed) knowledge of AI is much lower as compared to their approval, which
is, by contrast, quite high with respect to both AI generally considered and
several domain applications. This confirms other studies that observed a gap
between people’s limited competency and their perceptions and expectations,
which might be influenced by the narratives about the future progress of
emerging technologies such as AI [59].

#####  AI for the environment _vs_ The environmental impact of AI.

AI approval is often dependent on the sector or context of application, such
as education or healthcare. In this respect, the high acceptance rate of AI in
law enforcement and the environment is rather striking. A plausible
interpretation might be that people consider these as critical areas where the
use of advanced technologies, like AI, could ensure greater progress as
compared to other sectors. However, it is surprising that only a small portion
of the respondents choose societal and environmental aspects as one of their
ethical priorities. In other words, it seems that the intuition of the
beneficial effect of AI on important environmental challenges ahead is not on
par with the knowledge of possible negative impacts that AI may have on
society and the environment. This intuition would be in line with previous
studies showing that people tend to not care about the environmental impact of
AI solutions and pay more attention to transparency and explainability [47].

#####  Perceived AI impact _vs_ Knowledge about EU measures on Trustworthy AI.

While the perceived impact of AI is high across the interviewed population,
the knowledge of recent measures put forward by the EC to safeguard the risks
associated with the use of AI is significantly low. In particular, about 70%
of the respondents claim no knowledge about two key recent actions by the EC,
i.e. the ethics guidelines for Trustworthy AI [35] and the proposal for an AI
regulation [22], whereas most of them are familiar with the GDPR. Though this
lack of knowledge can be partially explained by the novelty of these
initiatives (April 2019 and April 2021 respectively), it seems that the public
discussion of the AI impact in the EU is still remote from citizens’
experience. Also, the lack of knowledge about the proposal for an EU
regulation on AI is somewhat in contrast with respondents’ policy preferences
indicating the introduction of laws as a top priority.

#####  Introduction of laws by national authorities _vs_ Trust in national
governments.

As anticipated, the set up of laws by national authorities is acknowledged as
(very) important by the largest portion of the respondents. However, national
governments are the second last entity that can be trusted a lot or somewhat.
This last opinion may reflect a larger discontent with democratic processes
[13], challenged by global crises (e.g. climate, migration, economy etc) and
more recently by the Covid-19 pandemic. The EU took a leading position in
proposing global standards for the governance of AI and promoting a unified
approach to AI across all member states. However, the implementation of these
policy and regulatory efforts might be undermined by the fragile relations
between citizens and democratic institutions and associated phenomena (e.g.
anti-EU sentiments and populist movements).

#####  AI Education as measure to improve citizens’ trust _vs_ Interest in
engaging with AI education.

With respect to the role of education in fostering trust in AI, 71%
respondents are highly positive and express a (strong) approval. The value of
education and culture is also reflected by the choice of universities and
research centers as the most trusted entities in ensuring the beneficial
development of AI. To gain a better understating of the value of education we
also asked participants if they would be interested in attending a free course
on AI with a view to improve their knowledge (see the last question, Q16).
Overall, 61% of participants answered positively, although compared to their
strong support of education-related initiatives, even higher percentages could
be expected. Moreover, only half of those who self-reported a low AI
competence (Q1) said they would be interested in attending a free course
(Q16). This limited interest in engaging with AI education might be indicative
of a sort of hesitancy in joining the innovation process brought about by AI,
in particular among individuals who feel less competent. A similar
interpretation may also apply to the selection of inclusive design teams and
consultation with stakeholders (Q12_6) as the least valued measures.

###  3.2 Potential barriers and recommendations

The combination of the implicit contradictions presented above suggests three
interrelated social trends. These may affect the way in which AI innovations
integrate into the fabric of social life and create a barrier to the human-
centric approach that the EU wishes to achieve. For each trend, we discuss
critical issues that policy makers could face and suggest a few
recommendations. We recall that the European AI strategy pursues three
fundamental goals: 1) boosting the AI uptake across the economy by private and
public sectors; 2) preparing for socio-economic changes brought by AI
transformations; 3) ensuring appropriate ethical and legal framework to
promote trustworthy AI [18].

####  3.2.1 Approval of a hyped, but poorly known, tech.

The divide between knowledge and approval, regardless of its causes, calls for
reflection on the meaning and implications of approving something which is not
sufficiently known or understood. Over the last few years we have witnessed an
explosion of fictional and non-fictional AI-related communication and
narratives. This large availability of information sources can contribute to
creating big expectations, on the one hand, but can also increase confusion or
even resistance and aversion [53, 15], on the other hand. For example, [25]
analyzed trends in beliefs, interests, and sentiment articles around AI in a
30-years period. Results show a significant increase in content with a
generally optimistic perspective since 2009, although certain topics regarding
ethical, technical, and social aspects of AI are also gaining relevance.
Moreover, the language used to communicate is highly influential [59]; when
mixed with fictional, or utopian narratives, it can create confusion and lead
the general public to overestimate the real capabilities and limitations of
AI, augmenting the disconnect from the real progress of the technology.

A manifest example of the risk of this poorly informed approval is the
attraction created by the latest language model ChatGPT. It seems plausible
that, in the imagination of a non-specialist, an AI system of this kind, which
creates poems, codes, and answers complex questions in a credible manner, is
likely to be credited with advanced cognitive abilities. The problem is that
systems like ChatGPT “can fool us into thinking that they understand more than
they do” [9], and this limitation is probably unknown by the majority of
users. The language and terminology used are fundamental to avoid inaccurate
and biased messages that create overhype and misinformation about the real
capabilities, limitations and associated risks of AI. Moreover, information
needs to be clear and adapted to the audience. To improve media communication
on AI and support more informed opinion we recommend: 1) increasing the study
of media communication on AI and social dynamics created by AI-related
content; 2) fostering training of science and tech journalists/communicators
on AI applications, in particular, on new AI breakthrough; 3) distributing
high-quality information through institutional channels (e.g. curating the
terminology and translating material in national languages).

####  3.2.2 Disconnect from public AI policies.

In democratic societies, institutions play a crucial role in anticipating
risks and taking preventive actions to protect citizens’ rights when
innovation processes take place. This is particularly important in times of
global crisis or rapid changes and when parts of the population lack the
expertise to face complex challenges [69]. However, the development and
implementation of public policies and laws are more effective when citizens
participate in the public discussion and gain a better understanding of the
issues at stake [27]. Indeed, increasing public awareness may impact people’s
values and priorities. Not surprisingly, privacy and data protection, which
turned out to be the most well-known EU action (i.e. the GDPR), is one of the
highest-rated ethical requirements by the respondents of the survey. We recall
that before the GDPR was released there were already a directive and
respective national laws regarding data privacy. Moreover, the regulation was
accompanied by a large campaign of information and awareness towards the
topic, in addition to a two-year adoption period for companies (from 2016 to
2018).

The implicit contrasts observed in our results stress the need of supporting
European citizens in gaining a greater understating of the risks associated
with AI, including harms that might be invisible to them. In particular, more
efforts should be made to raise awareness of AI’s environmental costs in the
public discourse as suggested by [6]. A poor understanding of societal harms
associated with AI may contribute to exacerbating inequalities and eroding
democratic processes. Moreover, if people have limited knowledge about the
rules and the initiatives introduced to protect them from potential AI-related
risks, they will not be aware of the rights they have and when these are
violated. Overall, reflecting on the gap between citizens and the EU policy
efforts on AI stresses the importance of building a culture of trust on top of
laws and policies. Educational and dissemination resources are needed to
promote the last key EU policy initiatives and to empower citizens to know
their rights and exercise them. Greater attention should also be directed to
the initiatives of inclusive governance to avoid the so-called paradox of
participation [11, 58], i.e. inclusion processes failing to achieve structural
reforms. To improve participation and make society a relevant stakeholder in
AI policy-making we recommend 1) analyzing the effectiveness of EU initiatives
and platforms aimed at stakeholders’ participation, including the European AI
alliance [1]; 2) creating information material on the AI-related risks and
associated EU measures targeted to different audiences (e.g. children and
seniors); 3) increasing local initiatives (including physical events) on AI
and the EU efforts aimed at reaching segments of society who are at the edge
of current AI debates (e.g because lacking technology or other cultural
resources).

####  3.2.3 Poor engagement with AI education and training.

Education and lifelong learning play a central role in the European AI policy.
These strategies aim at boosting economic growth but also preparing the
society as a whole, to ensure that “none is left behind in the digital
transformation” [18]. This preparation includes the introduction of AI from
the early stages of education to increase skilled workers in AI-related tasks,
but also the promotion of conditions that make Europe able to attract and
retain talent for AI research and industry. These steps connect with the need
to preserve democracy and core values in our society increasingly shaped by
AI, big data, and behavioral economics [34, 14]. As well as the Digital Europe
Programme (DIGITAL), the EU supports several projects to train AI experts and
stimulate excellence (e.g. TAILOR, ELISE, HumanE-AI-Net). European countries
are also making efforts to improve AI education at a national level as
reported in their AI strategies [26] and the AI Watch investment dashboard -
apparently the investments made in talent, skills, and lifelong learning
represent about 60% of total investments by private and public organizations
[2].

While it is widely acknowledged that education and training are key in
promoting citizens’ participation, what such an education should look like is
open to discussion. This problem regards the type of knowledge and skills that
we will value in the future. As the economy increasingly relies on AI, we
expect that AI-related skills, such as algorithmic formalism [32], will take a
greater role in education and culture. However, this change may favor critical
processes such as the prioritization of algorithmic thinking over other forms
of knowledge [5] and the subordination of education to business and economic
interests. Note that the influence of economic drivers in the shaping of AI
education could also damage the very field of AI by increasing the role of
techniques and approaches with a higher economic and commercial impact and
marginalizing the others. Another issue regards how to deliver AI education
and training. Several resources are available online supporting self-education
on AI, many offered for free [16]. However, if this becomes the default
option, some people might be excluded from AI education, such as workers who
have a low level of formal education and digital skills. We should also
consider to what extent people feel comfortable with the education offered,
whether they experience anxiety or social pressure. Further concerns regard
courses offered by big tech companies and how these can influence the public
discourse on AI as well as AI research [54].

To address the issues connected to the shaping of AI education we recommend 1)
assessing to what extent people feel conformable with existing educational
resources on AI and identifying categories of the population that might be
excluded; 2) increasing the integration of the humanities into computer
science and AI curricula to help future tech people address broader socio-
technical challenges; 3) reconsidering the incentives of research careers, now
dictated by the dynamics and standards of individual disciplines, in light of
multidisciplinary collaborations and societal challenges raised by techno-
science.

##  4 Conclusions

This paper presents and discusses the results obtained from the PAICE
questionnaire. The collected responses show that European citizens have low
knowledge of AI capabilities in different applications and domains, as well as
of the efforts aimed to build an ethical and regulatory framework for this
technology. The analysis of our results suggests some tensions connected to
broader social trends that lead to reflection on aspects that may interfere
with policy efforts towards Trustworthy AI: 1) an uninformed approval recalls
attention to the risks of misinformation and poor narratives about AI; 2) a
disconnect from EU policy on AI brings attention to the need of high-quality
communication campaigns on the AI-related harms and current EU policy and
regulatory efforts; 3) a poor engagement with AI education and training
strategies points to the risks of growing social and cultural inequalities.

Through the analysis of the validity and reliability of the questionnaire
(PAICE) we assess the robustness of the theoretical structure identified by
the working group during the design process and support the research community
in the reuse of the PAICE. The validation of the questionnaire shows that for
a subset of items, PAICE can be used to measure awareness, attitude, and trust
towards the AI ecosystem. In addition, PAICE proves useful in providing
respondents with new stimuli making them reflect upon their interaction with
new technologies and its possible impact on society. At the end of the
questionnaire, we repeated item Q3 investigating the perceived impact of AI,
and found that 62.2% of respondents answered that AI has an impact on their
daily life, an increase of 10 percentage points. In future work, we plan to
extend the questionnaire to new regions and investigate country-specific
differences with available data on the AI landscape [39].

##  5 Supplementary Material

In the following subsections, we provide the links to the supplemental
material of the present research work.

###  5.1 Questionnaire

Text in full of the questionnaire on the Perceptions of AI by the Citizens of
Europe (PAICE) translated in English: https://github.com/EU-
Survey/Material/blob/main/S1_quest.pdf

###  5.2 Responses

Table with all aggregated responses to likert-scale / dichotomous / multi-
response items and rankings: https://github.com/EU-
Survey/Material/blob/main/S2_res.xlsx. For likert scale items, some
descriptive statistics are also reported.

###  5.3 Digital skills

Table with aggregated responses related to digital skills, education and
population size grouped by countries: https://github.com/EU-
Survey/Material/blob/main/S3_dem.xlsx

###  5.4 Statistics by Groups

Table with all responses to likert scale items aggregated by countries / age
groups / gender with p-values: https://github.com/EU-
Survey/Material/blob/main/S4_comp.xlsx Responses are presented with respect to
the dimension considered (awareness, attitude, and trust).

###  5.5 Exploratory Factor Analysis: Table

Table with the results of the Exploratory Factor Analysis:
https://github.com/EU-Survey/Material/blob/main/S5_efa.xlsx. Exploratory
factor analysis (EFA) is based on the polychoric matrix which uses principal
axis factoring with oblique rotation

###  5.6 Exploratory Factor Analysis: Figure

Plot of Exploratory Factor Analysis: https://github.com/EU-
Survey/Material/blob/main/S6_efa.pdf

## References

  * [1] The european ai alliance, 2018.
  * [2] Ai investments dashboard, 2018.
  * [3] Michele Biasutti and Sara Frate.  A validity and reliability study of the attitudes toward sustainable development scale.  Environmental Education Research, 23(2):214–230, 2017.
  * [4] Miranda Bogen and Aaron Rieke.  Help wanted: An examination of hiring algorithms, equity, and bias.  Technical report, Upturn, 2018.
  * [5] Danah Boyd and Kate Crawford.  Critical questions for big data: Provocations for a cultural, technological, and scholarly phenomenon.  Information, communication & society, 15(5):662–679, 2012.
  * [6] Benedetta Brevini.  Black boxes, not green: Mythologizing artificial intelligence and omitting the environment.  Big Data & Society, 7(2):2053951720935141, 2020.
  * [7] Timothy A Brown.  Confirmatory factor analysis for applied research.  Guilford publications, 2015.
  * [8] Stephen Cave, Kate Coughlan, and Kanta Dihal.  ”scary robots”: Examining public responses to ai.  In Proceedings of the 2019 AAAI/ACM Conference on AI, Ethics, and Society, AIES ’19, page 331–337, New York, NY, USA, 2019. Association for Computing Machinery.
  * [9] Brian X. Chen.  How to use chatgpt and still be a good person, 2022.
  * [10] Jing Cheng and Jinghan Zeng.  Shaping ai’s future? china in global ai governance.  Journal of Contemporary China, pages 1–17, 2022.
  * [11] Frances Cleaver.  Paradoxes of participation: questioning participatory approaches to development.  Journal of International Development: The Journal of the Development Studies Association, 11(4):597–612, 1999.
  * [12] L. Cohen, L. Manion, and K. Morrison.  Research Methods in Education 8 edition.  Routledge, Milton Park, AB, USA, 2017.
  * [13] Challenges to democracy in europe: Insights into a complex and turbulent political climate, 2021.
  * [14] Nello Cristianini and Teresa Scantamburlo.  On social machines for algorithmic regulation.  AI & SOCIETY, 35:645–662, 2020.
  * [15] Berkeley J Dietvorst, Joseph P Simmons, and Cade Massey.  Algorithm aversion: people erroneously avoid algorithms after seeing them err.  Journal of Experimental Psychology: General, 144(1):114, 2015.
  * [16] Elements of ai, 2018.
  * [17] European Commission.  Special eurobarometer 460-attitudes towards the impact of digitisation and automation on daily life.  2017\.
  * [18] European Commission.  Communication from the commission to the european parliament, the european council, the council, the european economic and social committee and the committee of the regions. artificial intelligence for europe, 2018.
  * [19] European Commission.  Standard eurobarometer 92. europeans and artificial intelligence.  2019\.
  * [20] European Commission.  White Paper on Artificial Intelligence - A European Approach to Excellence and Trust, 2019.
  * [21] European Commission.  Special eurobarometer 516-european citizens’ knowledge and attitudes towards science and technology.  2021\.
  * [22] European Parliament.  Proposal for a regulation of the european parliament and of the council laying down harmonised rules on artificial intelligence (artificial intelligence act) and amending certain union legislative acts, 2021.
  * [23] Eurostat.  Educational attainment statistics, 2021.
  * [24] Leandre R Fabrigar, Duane T Wegener, Robert C MacCallum, and Erin J Strahan.  Evaluating the use of exploratory factor analysis in psychological research.  Psychological methods, 4(3):272, 1999.
  * [25] Ethan Fast and Eric Horvitz.  Long-term trends in the public perception of artificial intelligence.  In Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence, AAAI’17, page 963–969. AAAI Press, 2017.
  * [26] Francesca Foffano, Teresa Scantamburlo, and Atia Cortés.  Investing in ai for social good: An analysis of european national strategies.  AI & Society, 2022.
  * [27] William Funk.  Public participation and transparency in administrative law-three examples as an object lesson.  Admin. L. Rev., 61:171, 2009.
  * [28] Anne M Gadermann, Martin Guhn, and Bruno D Zumbo.  Estimating ordinal reliability for likert-type and ordinal item response data: A conceptual, empirical, and practical guide.  Practical Assessment, Research, and Evaluation, 17(1):3, 2012.
  * [29] Felix Gille, Anna Jobin, and Marcello Ienca.  What we talk about when we talk about trust: Theory of trust for ai in healthcare.  Intelligence-Based Medicine, 1:100001, 2020.
  * [30] Ella Glikson and Anita Williams Woolley.  Human trust in artificial intelligence: Review of empirical research.  Academy of Management Annals, 14(2):627–660, 2020.
  * [31] South African Government.  Presidential commission on fourth industrial revolution, 2019.
  * [32] Ben Green and Salomé Viljoen.  Algorithmic realism: expanding the boundaries of algorithmic thought.  In Proceedings of the 2020 conference on fairness, accountability, and transparency, pages 19–31, 2020.
  * [33] Thilo Hagendorff.  The ethics of ai ethics: An evaluation of guidelines.  Minds and Machines, 30(1):99–120, 2020.
  * [34] Dirk Helbing, Bruno S Frey, Gerd Gigerenzer, Ernst Hafen, Michael Hagner, Yvonne Hofstetter, Jeroen Van Den Hoven, Roberto V Zicari, and Andrej Zwitter.  Will democracy survive big data and artificial intelligence? Springer, 2019.
  * [35] High-Level Expert Group on Artificial Intelligence.  Ethics Guidelines for Trustworthy AI, 2019.
  * [36] Soraj Hongladarom and Jerd Bandasak.  Non-western ai ethics guidelines: implications for intercultural ethics of technology.  AI & Society, 2023.
  * [37] Alon Jacovi, Ana Marasović, Tim Miller, and Yoav Goldberg.  Formalizing trust in artificial intelligence: Prerequisites, causes and goals of human trust in ai.  In Proceedings of the 2021 ACM conference on fairness, accountability, and transparency, pages 624–635, 2021.
  * [38] Maurice Jakesch, Zana Buçinca, Saleema Amershi, and Alexandra Olteanu.  How different groups prioritize ethical values for responsible ai.  In 2022 ACM Conference on Fairness, Accountability, and Transparency, pages 310–323, 2022.
  * [39] Joint Research Centre.  Ai watch index 2021, 2022.
  * [40] K.G Jöreskog.  On the estimation of polychoric correlations and their asymptotic covariance matrix.  Psychometrika, 59:381–389, 2016.
  * [41] Ankur Joshi, Saket Kale, Satish Chandel, and D Kumar Pal.  Likert scale: Explored and explained.  British journal of applied science & technology, 7(4):396, 2015\.
  * [42] Henry F Kaiser.  An index of factorial simplicity.  Psychometrika, 39(1):31–36, 1974.
  * [43] Patrick Gage Kelley, Yongwei Yang, Courtney Heldreth, Christopher Moessner, Aaron Sedley, Andreas Kramm, David T. Newman, and Allison Woodruff.  Exciting, useful, worrying, futuristic: Public perception of artificial intelligence in 8 countries.  In Proceedings of the 2021 AAAI/ACM Conference on AI, Ethics, and Society, AIES ’21, page 627–637, New York, NY, USA, 2021. Association for Computing Machinery.
  * [44] Aphra Kerr, Marguerite Barry, and John D Kelleher.  Expectations of artificial intelligence and the performativity of ethics: Implications for communication governance.  Big Data & Society, 7(1):2053951720915939, 2020.
  * [45] Zafar A Khan and Dilan Jayaweera.  Smart meter data based load forecasting and demand side management in distribution networks with embedded pv systems.  IEEE Access, 8:2631–2644, 2019.
  * [46] Kimon Kieslich, Birte Keller, and Christopher Starke.  Artificial intelligence ethics by design. evaluating public perception on the importance of ethical design principles of artificial intelligence.  Big Data & Society, 9(1):20539517221092956, 2022.
  * [47] Pascal D König, Stefan Wurster, and Markus B Siewert.  Consumers are willing to pay a price for explainable, but not for green ai. evidence from a choice-based conjoint analysis.  Big Data & Society, 9(1):20539517211069632, 2022.
  * [48] Troels Krarup and Maja Horst.  European artificial intelligence policy as digital single market making.  Big Data & Society, 10(1):20539517231153811, 2023.
  * [49] Min Kyung Lee.  Understanding perception of algorithmic decisions: Fairness, trust, and emotion in response to algorithmic management.  Big Data & Society, 5(1):2053951718756684, 2018.
  * [50] CH Li.  Confirmatory factor analysis with ordinal data: Comparing robust maximum likelihood and diagonally weighted least squares.  Behavior Research Methods volume, 48:936–949, 2016.
  * [51] Rensis Likert.  A Technique for the Measurement of Attitudes.  New York: Columbia University Press, 1932.
  * [52] Steven Lockey, Nicole Gillespie, Daniel Holm, and Ida Asadi Someh.  A review of trust in artificial intelligence: Challenges, vulnerabilities and future directions.  2021\.
  * [53] Chiara Longoni, Andrea Bonezzi, and Carey K Morewedge.  Resistance to medical artificial intelligence.  Journal of Consumer Research, 46(4):629–650, 2019.
  * [54] Inga Luchs, Clemens Apprich, and Marcel Broersma.  Learning machine learning: On the political economy of big tech’s online ai courses.  Big Data & Society, 10(1):20539517231153806, 2023.
  * [55] Marketing problem solving srl, 2021.
  * [56] Jum C Nunnally.  Psychometric theory 3E.  Tata McGraw-hill education, 1994.
  * [57] The oecd ai policy observatory, 2019.
  * [58] Marie-Therese Png.  At the tensions of south and north: Critical roles of global south stakeholders in ai governance.  In 2022 ACM Conference on Fairness, Accountability, and Transparency, pages 1434–1445, 2022.
  * [59] The Royal Society.  Portrayals and perceptions of AI and why they matter.  Technical report, 2018.
  * [60] Laura Sartori and Giulia Bocca.  Minding the gap (s): public perceptions of ai and socio-technical imaginaries.  AI & SOCIETY, pages 1–16, 2022.
  * [61] Keng Siau and Weiyu Wang.  Building trust in artificial intelligence, machine learning, and robotics.  Cutter business technology journal, 31(2):47–53, 2018.
  * [62] Katharina Simbeck.  Facct-check on ai regulation: Systematic evaluation of ai regulation on the example of the legislation on the use of ai in the public sector in the german federal state of schleswig-holstein.  In 2022 ACM Conference on Fairness, Accountability, and Transparency, pages 89–96, 2022.
  * [63] Gita Taasoobshirazi and Shanshan Wang.  The performance of the srmr, rmsea, cfi, and tli: An examination of sample size, path size, and degrees of freedom.  Journal of Applied Quantitative Methods, 11(3):31–39, 2016.
  * [64] H. E. Tinsley and S. D. Brown.  Research Methods in Education 8 edition.  London: Academic Press, 2020.
  * [65] US Congress.  Text - h.r.6580 - 117th congress (2021-2022): Algorithmic accountability act of 2022, 2022.
  * [66] Baobao Zhang and Allan Dafoe.  Artificial intelligence: American attitudes and trends.  Available at SSRN 3312874, 2019.
  * [67] Baobao Zhang and Allan Dafoe.  U.s. public opinion on the governance of artificial intelligence.  In Proceedings of the AAAI/ACM Conference on AI, Ethics, and Society, AIES ’20, page 187–193, New York, NY, USA, 2020. Association for Computing Machinery.
  * [68] Bruno D Zumbo, Anne M Gadermann, and Cornelia Zeisser.  Ordinal versions of coefficients alpha and theta for likert rating scales.  Journal of modern applied statistical methods, 6(1):4, 2007.
  * [69] Andrej Zwitter.  The rule of law in times of crisis: A legal theory on the state of emergency in the liberal democracy.  ARSP: Archiv Für Rechts-Und Sozialphilosophie/Archives for Philosophy of Law and Social Philosophy, pages 95–111, 2012.


# Finite groups with integer harmonic mean of element orders

Iulia Cătălina Pleşca111 Faculty of Mathematics of ”Al. I. Cuza” University of
Iaşi, Romania, e-mail<EMAIL_ADDRESS>ORCID: 0000-0001-7140-844X and
Marius Tărnăuceanu222Faculty of Mathematics of ”Al. I. Cuza” University of
Iaşi, Romania, e-mail<EMAIL_ADDRESS>
(September 9, 2023)

###### Abstract

In this paper, we introduce a new function computing the harmonic mean of
element orders of a finite group. We present a series of properties for this
function, and then we study groups for which the value of the function is an
integer.

###### Acknowledgements

The authors are grateful to the reviewer for their remarks which improved the
previous version of the paper.

###### Funding

The authors did not receive support from any organization for the submitted
work.

###### Conflicts of interests

The authors declare that they have no conflict of interest.

##  1 Introduction

Throughout the article, let $G$ be a finite group and denote by $o(a)$ the
order of an element $a\in G$.

In the last three decades, an increasing number of functions and concepts from
number theory have been adapted to group theory. For example, Leinster
introduced what would later be called Leinster groups [8], a group analogue of
perfect numbers. Subsequently, a plethora of arithmetic functions that involve
the orders of elements/subgroups of groups have been introduced. We refer the
reader to [5] for a recent survey on this topic.

Special attention has also been given to various means of element orders:
arithmetic in [7, 4] and geometric in [4]. In our paper, we will focus on the
harmonic mean. Let us recall the functions that represent the starting point
for our work.

Two of the most common functions in number theory associated to a positive
integer $n\in\mathbb{N}^{*}$ are the number of its divisors $\tau(n)$ and the
sum of its divisors $\sigma(n)$. These two functions have been adapted to
group theory as follows. Given a finite group, $G$, $\tau(G)$ represents the
number of its normal subgroups and $\sigma(G)$ the sum of their orders.

These latter two appear in the definition of a new function

$H(G)=|G|\frac{\tau(G)}{\sigma(G)}\,,$

introduced in [2] that was meant to generalize the arithmetic function
$H(n)=n\tau(n)/\sigma(n)$.

The last function that inspired us, the sum of inverses of the orders of
elements

$m(G)=\sum_{a\in G}\frac{1}{o(a)}\,,$

was introduced in [1] and, similarly to other sum functions, can give
characterizations for commutativity, ciclicity and nilpotency.

Inspired by these, we introduce the function “the harmonic mean of element
orders of a finite group”, i.e.

$h_{m}(G)=\frac{|G|}{m(G)}\,.$ (1)

Throughout this article we put a dent in the following problem:

######  Question 1.

Which are the finite groups $G$ with $h_{m}(G)\in\mathbb{N}$?

We start by giving some immediate properties of the newly introduced function
including a lower bound that is reached only by finite $p$-groups. We continue
by characterizing the finite $p$-groups with integer $h_{m}$. In the end, we
observe that $h_{m}^{-1}(2)=\\{C_{4},D_{8}\\}$ and study $h_{m}^{-1}(3)$.

We use the following notations for some of the most common groups. For
$m,n\in\mathbb{N}^{*}$, $C_{n}$ denotes the cyclic group of order $n$, $D_{n}$
the dihedral group of order $n$, $S_{n}$ the symmetric group of order $n$,
$SD_{n}$ the semidihedral group of order $n$, $SL(m,n)$ the special linear
group of degree $m$ over a field of $n$ elements. In addition, $Q_{8}$ is the
quaternion group. The rest of the notations are standard. Basic notions and
results on groups can be found in [6].

##  2 Main results

A series of inequalities for the function $m$ are given in [1] (Lemmas 1.3,
2.2, 2.3, 2.4, 2.6). We summarize these in the first proposition:

######  Proposition 2.1 ([1]).

The following properties hold for the function $m$:

  1. a)

If $|G|=n$, then $m(C_{n})\leq m(G)$, with equality if and only if $G\cong
C_{n}$.

  2. b)

If $H\leq G$, then $m(H)\leq m(G)$, with equality if and only if $H=G$.

  3. c)

If $N\unlhd G$, then $m(G/N)\leq m(G)$, with equality if and only if $N=1$.

  4. d)

If $P$ is a normal cyclic Sylow $p$-subgroup of $G$:
$G\in\operatorname{Syl}_{p}(G)$, then $m(Px)\geq m(P)/o(Px)$, where $Px\in
G/P$. Equality holds if and only if x centralizes $P$. Also, $m(G)\geq
m(P)m(G/P)$ with equality if and only if $P$ is central in $G$.

  5. e)

If $G_{1},G_{2}$ are finite groups, then $m(G_{1}\times G_{2})\geq
m(G_{1})m(G_{2})$, with equality if and only if the orders of the groups are
coprime: $\gcd(|G_{1}|,|G_{2}|)=1$.

We deduce a corresponding proposition for $h_{m}$:

######  Proposition 2.2.

The following properties hold:

  1. a)

If $|G|=n$, then $h_{m}(G)\leq h_{m}(C_{n})$. Equality holds if and only if
$G\cong C_{n}$.

  2. b)

For a subgroup $H\leq G$, it follows that $h_{m}(G)\leq[G:H]h_{m}(H)$.
Equality holds if and only if $H=G$.

  3. c)

For a normal subgroup $H\lhd G$, it follows that
$h_{m}(G)\leq|H|h_{m}(\frac{G}{H})$. Equality holds if and only if $H$ is the
trivial subgroup.

  4. d)

If $P$ is a normal cyclic Sylow $p$-subgroup of $G$:
$G\in\operatorname{Syl}_{p}(G)$, then $h_{m}(G)\leq
h_{m}(P)h_{m}(\frac{G}{P})$. Equality holds if and only if $P$ is central in
$G$.

  5. e)

$h_{m}$ is multiplicative: for all finite groups $G_{1},G_{2}$ of coprime
orders, we have $h_{m}(G_{1}\times G_{2})=h_{m}(G_{1})h_{m}(G_{2})$. This
shows that the study of the function $h_{m}$ for finite nilpotent groups can
be reduced to $p$-groups.

Throughout this study, a lower bound for the function $h_{m}$ is needed and
can be obtained from the following lemma.

######  Lemma 2.3.

Let $G$ be a finite group, $C(G)=\\{H\leq G\mid H\text{ cyclic}\\}$ and $p$
the smallest prime divisor of $|G|$. Then the following inequality holds:

$h_{m}(G)\geq\frac{p|G|}{(p-1)|C(G)|+1}.$ (2)

Equality holds if and only if $G$ is a $p$-group.

###### Proof.

Let $d_{1}=1,d_{2}=p,\dots,d_{r}$ be the orders of the elements in $G$.

For all $i\in\overline{1,r}$, we introduce the following notations:

$n_{i}=|\\{a\in G\mid o(a)=d_{i}\\}|\mbox{ and }n_{i}^{\prime}=|\\{H\in
C(G)\mid|H|=d_{i}\\}|.$

Then the following relations hold:

$\displaystyle m(G)$
$\displaystyle=\sum_{i=1}^{r}\frac{n_{i}}{d_{i}}=\sum_{i=1}^{r}n_{i}^{\prime}\frac{\varphi(d_{i})}{d_{i}}=1+\sum_{i=2}^{r}n_{i}^{\prime}\frac{\varphi(d_{i})}{d_{i}}$
$\displaystyle=1+\sum_{i=2}^{r}n_{i}^{\prime}\prod_{q|d_{i},q\text{
prime}}\left(1-\frac{1}{q}\right)$ $\displaystyle\leq
1+\frac{p-1}{p}(n_{2}^{\prime}+\dots+n_{r}^{\prime})=\frac{(p-1)|C(G)|+1}{p}\,,$

which give (2).

The equality case happens if and only if $p$ is the only prime divisor of
$|G|$, which holds if and only if $G$ is a $p$-group. ∎

######  Remark.

Since $|C(G)|\leq|G|$, (2) gives the anticipated lower boundary for $h_{m}$ in
terms of the smallest prime divisor $p$ of $|G|$:

$h_{m}(G)\geq\frac{p|G|}{(p-1)|G|+1}\,.$

For finite $p$-groups $G$ we can establish when $h_{m}(G)\in\mathbb{N}$
through the following result.

######  Theorem 2.4.

Let $G$ be a finite $p$-group. Then $h_{m}(G)\in\mathbb{N}$ if and only if $G$
is cyclic of order $p^{\sum_{i=1}^{s}p^{i}}$, with $s\in\mathbb{N}^{*}$, or
$G\simeq D_{8}$.

###### Proof.

Let $|G|=p^{n}$, $\exp(G)=p^{m}$ and

$n_{i}=|\\{a\in G\mid o(a)=p^{i}\\}|\mbox{ and }n_{i}^{\prime}=|\\{H\in
C(G)\mid|H|=p^{i}\\}|,\,\forall i\in\overline{0,m}.$

Lemma 2.3 implies that

$h_{m}(G)=\frac{p^{n+1}}{(p-1)|C(G)|+1}\,,$

which leads to the following equivalences:

$\displaystyle h_{m}(G)\in\mathbb{N}$ $\displaystyle\text{ if and only if
}(p-1)|C(G)|+1=p^{t}\text{ with }t\leq n+1$ (3) $\displaystyle\text{ if and
only if }|C(G)|=p^{t-1}+p^{t-2}+\dots+p+1,\text{ with }t\leq n.$

For $p=2$, this formula becomes

$|C(G)|=2^{t}-1.$ (4)

Going back to the proof, we distinguish the following two cases:

Case 1: $p$

odd. There are two possibilities: $G$ can be cyclic or not.

  1. a)

Suppose $G$ is not cyclic

Using Theorem 1.10 from [3], it follows that
$\begin{cases}n_{1}^{\prime}\equiv p+1\pmod{p^{2}}\\\
n_{2}^{\prime},\dots,n_{m}^{\prime}\equiv 0\pmod{p}\end{cases}$ and so,
recalling that the trivial subgroup is an element of $C(G)$, we obtain
$|C(G)|\equiv 2\pmod{p}$. Thus there are no solutions.

  2. b)

Suppose $G$ is cyclic

From (3), it follows that $n+1=p^{t-1}+p^{t-2}+\dots+p+1,\text{ with }t\leq
n\xRightarrow{s=t-1}n=\sum_{i=1}^{s}p^{i}.$

Case 2: $p=2$

  1. a)

Suppose $G$ is not cyclic and $G$ is not of maximal class

Using Theorem 1.17 from [3], it follows that
$\begin{cases}n_{1}^{\prime}\equiv 3\pmod{4}\\\
n_{2}^{\prime},\dots,n_{m}^{\prime}\equiv 0\pmod{2}\end{cases}$ and so
$|C(G)|\equiv 0\pmod{2}$. Thus there are no solutions.

  2. b)

Suppose $G$ is cyclic

Therefore we have $|C(G)|=n+1$. Using (4), it follows that
$n=2^{t}-2\xlongequal{s=t-1}\sum_{i=1}^{s}2^{i}$ with $1\leq s\leq n+1$.

  3. c)

Suppose $G$ is of maximal class

It follows that $G\in\\{D_{2^{n}},Q_{2^{n}},SD_{2^{n}}\\}$. Using [9], the
following analysis is obtained:

    1. i)

$G\cong D_{2^{n}}\Rightarrow|C(G)|=2^{n-1}+n\Rightarrow 2^{n-1}+n+1|2^{n+1}$.
It follows that the left hand side has to be a power of $2$:
$2^{n-1}+n+1=2^{t}$. This happens when $2^{n-1}=n+1$. Since
$2^{n-1}>n+1,\forall n>3$ (it can be shown inductively), it can be proven by
direct computation that the only solution is $n=3$ and therefore $G\cong
D_{8}$.

    2. ii)

$G\cong Q_{2^{n}}\Rightarrow|C(G)|=2^{n-2}+n\Rightarrow 2^{n-2}+n+1|2^{n+1}$.
Reasoning similarly as above, it follows that there are no solutions.

    3. iii)

$G\cong SD_{2^{n}}\Rightarrow|C(G)|=3\cdot 2^{n-3}+n\Rightarrow 3\cdot
2^{n-3}+n+1|2^{n+1}$. The left hand side must be a power of $2$: $3\cdot
2^{n-3}+n+1=2^{s}$, therefore $n+1$ is divisible by $2^{n-3}$. Using
$2^{n-3}>n+1,\forall n>5$ (it can be again shown inductively), it can be
proved that there are no solutions.

The proof of Theorem 2.3 is now complete. ∎

We note that Theorem 2.3 gives the following characterization for $D_{8}$.

######  Corollary 2.5.

$D_{8}$ is the only non-cyclic $p$-group with integer harmonic mean of element
orders.

An alternative characterization is the following:

######  Proposition 2.6.

$D_{8}$ is the only dihedral group with integer harmonic mean of element
orders.

###### Proof.

$\text{Let }D_{2n}\text{ be a dihedral group of order }2n\text{ such that
}h_{m}(D_{2n})\in\mathbb{N}^{*}.$ (5)

Let $n=p_{1}^{n_{1}}\cdot p_{2}^{n_{2}}\cdots p_{k}^{n_{k}}$ be the
decomposition of $n$ as a product of prime factors, where
$p_{1}<p_{2}<\dots<p_{k}$.

Let us write $D_{2n}=\\{1,r,\dots,r^{n-1},s,sr,\dots,sr^{n-1}\\}$, where $r$
is the rotation of order $\frac{2\pi}{n}$ and $s$ is the reflection around a
vertex and the center of a regular polygon with $n$-sides. Obviously, $\langle
r\rangle\cong C_{n}$ and $o(sr^{i})=2,\forall i\in\overline{0,n-1}$.

If we compute $h_{m}(D_{2n})$, we get

$h_{m}(D_{2n})=\frac{|G|}{\sum_{i=0}^{n-1}\frac{1}{o(r^{i})}+\sum_{i=0}^{n-1}\frac{1}{o(sr^{i})}}=\frac{2n}{m(C_{n})+\frac{n}{2}}.$

Let us denote

$\alpha:=\frac{2n}{m(C_{n})+\frac{n}{2}}\in\mathbb{N}^{*}$

because of assumption (5). Since $m(C_{n})>0$, it follows that $\alpha<4$ and
therefore three cases can occur:

  1. a)

$\alpha=1$

Then $m(C_{n})=\frac{3n}{2}$ , contradicting $m(C_{n})=\sum_{a\in
C_{n}}\frac{1}{o(a)}\leq\sum_{a\in C_{n}}1=n$.

  2. b)

$\alpha=2$

It follows that

$m(C_{n})=\frac{n}{2}.$ (6)

Clearly $C_{n}\cong\prod_{i=1}^{k}C_{p_{i}^{n_{i}}}$. According to Proposition
2.1 e), it follows that

$m(C_{n})=\prod_{i=1}^{k}m(C_{p_{i}^{n_{i}}}).$ (7)

Each factor in the right-hand side can be computed. Fix $i\in\overline{1,k}$.
For each $j\in\overline{1,n_{i}}$, there are $\varphi(p_{i}^{j})$ elements of
order $p_{i}^{j}$. Therefore $\displaystyle
m(C_{p_{i}^{n_{i}}})=1+\sum_{j=1}^{n_{i}}\frac{\varphi(p_{i}^{j})}{p_{i}^{j}}=1+\sum_{j=1}^{n_{i}}\frac{p_{i}^{j}(1-\frac{1}{p_{i}})}{p_{i}^{j}}=1+\sum_{j=1}^{n_{i}}\left(1-\frac{1}{p_{i}}\right)=\frac{p_{i}-1+1}{p_{i}}+\frac{n_{i}}{p_{i}}\left(p_{i}-1\right)=\frac{(n_{i}+1)(p_{i}-1)+1}{p_{i}}.$
Using (6) and (7), it follows that

$\prod_{i=1}^{k}\left((p_{i}-1)(n_{i}+1)+1\right)=\frac{1}{2}p_{1}^{n_{1}+1}\cdot\dots\cdot
p_{n}^{n_{k}+1}.$

Since the left-hand side is an integer, it follows that $p_{1}=2$, so

$(n_{1}+2)\prod_{i=2}^{k}((p_{i}-1)(n_{i}+1)+1)=2^{n_{1}}p_{2}^{n_{2}+1}\dots
p_{k}^{n_{k}+1}$

Bernoulli’s inequality gives:

$p_{i}^{n_{i}+1}=\left(1+(p_{i}-1)\right)^{n_{i}+1}>(p_{i}-1)(n_{i}+1)+1,\forall
i=\overline{2,k},$

therefore $n_{1}+2>2^{n_{1}}$, i.e. $n_{1}=1$.

If $k>1$ it follows that

$3\prod_{i=2}^{k}\left((p_{i}-1)(n_{i}+1)+1\right)=2p_{2}^{n_{2}+1}\cdot\dots\cdot
p_{k}^{n_{k}+1},$

therefore $p_{2}=3$ and

$3(2n_{2}+3)\prod_{i=3}^{k}\left((p_{i}-1)(n_{i}+1)+1\right)=2\cdot
3^{n_{2}+1}p_{3}^{n_{3}+1}\cdot\dots p_{k}^{n+1}.$

Then $2n_{2}+3\geq 2\cdot 3^{n_{2}}$, which does not yield solutions. Thus the
assumption that $k>1$ does not hold, and consequently the only solution is
$k=1$ and $n_{1}=2$, which gives $n=4$.

  3. c)

$\alpha=3$

The analysis is analogous to the previous case. Alternatively, the result
follows from Theorem 2.6.

∎

######  Remark.

Let us note that we can build nilpotent groups $G$ with
$h_{m}(G)\in\mathbb{N}$ as direct products of $p$-groups of the type of the
ones in Theorem 2.4.

In addition, we can construct non-nilpotent groups $G$ with this property, for
example $G=\operatorname{SL}(2,3)\times C_{7^{7}}$. The idea is to start with
a non-nilpotent group $G_{1}$ and to do a direct product with a cyclic group
$G_{2}$ such that the denominator of $h_{m}(G_{1})$ reduces. In the example
above:

$h_{m}(G)=h_{m}(\operatorname{SL}(2,3))\cdot
h_{m}(C_{7^{7}})=\frac{24}{7}\cdot 7^{6}=24\cdot 7^{5}.$

In what follows, we will study the integer values of the function $h_{m}$.

Obviously, we have $h_{m}(G)=1$ if and only if $G$ is the trivial group.

In addition, Proposition 2.6 and its proof give that
$h_{m}(C_{4})=h_{m}(D_{8})=2$. We can show that these are the only groups $G$
with $h_{m}(G)=2$.

######  Theorem 2.7.

Let $G$ be a finite group. Then $h_{m}(G)=2$ if and only if $G\cong C_{4}$ or
$G\cong D_{8}$.

###### Proof.

Since $h_{m}(G)=2$, it follows that $2||G|$, therefore (2) becomes

$2\geq\frac{2|G|}{|C(G)|+1}\,,$

i.e.

$|C(G)|\geq|G|-1.$ (8)

Using the same notations as in Lemma 2.3, it follows that $d_{2}=2$ and

$|G|=\sum_{i=1}^{r}n_{i}=\sum_{i=1}^{r}n_{i}^{\prime}\varphi(d_{i})$
$|C(G)|=\sum_{i=1}^{r}n_{i}^{\prime},$

therefore (8) becomes

$\sum_{i=1}^{r}n_{i}^{\prime}(\varphi(d_{i})-1)\leq 1.$ (9)

Since $\varphi(d_{i})>1$, for $d_{i}>2$, we identify two possible cases:

  1. a)

$r=2$

It follows that $G$ is an elementary abelian $2$-group. Then
$h_{m}(G)\not\in\mathbb{N}$ by Theorem 2.4, which is a contradiction.

  2. b)

$r=3$

Clearly, we have $n_{3}^{\prime}=1$ and $\varphi(d_{3})=2$, and so $d_{3}=3$
or $d_{3}=4$.

If $d_{3}=3$, then $G$ has $1$ element of order $1$, $2$ elements of order $3$
and $|G|-3$ elements of order $2$. Thus
$h_{m}(G)=\frac{6|G|}{3|G|+1}\notin\mathbb{N}$, which is a contradiction.

If $d_{3}=4$, then $h_{m}(G)=2$. This means that $G$ is a $2$-group of
exponent $4$ with a single cyclic subgroup of order $4$. Using Theorem 2.4, we
obtain $G\cong C_{4}$ or $G\cong D_{8}$, as desired.

∎

######  Theorem 2.8.

The finite non-trivial groups $G$ with $h_{m}(G)\leq 2$ are
$C_{2}^{n},n\in\mathbb{N},C_{3},S_{3},C_{4}$ and $D_{8}$.

###### Proof.

Using the same ideas and notations as in the proof of Theorem 2.7, we can
classify the finite groups $G$ with $h_{m}(G)\leq 2$. We identify the
following cases:

  1. a)

$|G|$ is odd

It follows that

$\displaystyle h_{m}(G)$
$\displaystyle=\frac{|G|}{m(G)}\geq\frac{|G|}{1+\frac{|G|-1}{p_{1}}}\geq\frac{|G|}{1+\frac{|G|-1}{3}}=\frac{3|G|}{|G|+2}\Rightarrow$
$\displaystyle\frac{3|G|}{|G|+2}\leq$ $\displaystyle 2\Rightarrow|G|\leq
4\Rightarrow|G|=3\Rightarrow G\cong C_{3}.$

  2. b)

$|G|$ is even

It follows that (8) holds. There are two possibilities:

    1. i)

$r=2$

Then $G\cong C_{2}^{n}$, $n\in\mathbb{N}$.

    2. ii)

$r=3$

Then there are two possibilities:

$\mathbf{d_{3}=3}$

Inequality (9) gives that $n_{3}^{\prime}=1$ which means that $G$ has a unique
cyclic subgroup of order $3$, let us denote this by $H$. Since $G$ does not
contain cyclic subgroups of order $>3$, it follows that $H$ is the only
$3$-Sylow subgroup of $G$. It is also normal. Since $|G|$ even, it follows
that there is also a $2$-Sylow subgroup of order $2$. Thus $G=HK$. Since $G$
does not have cyclic subgroups of order $6$, it follows that $C_{K}(H)=1$,
therefore $|K|=|{\rm Aut}(H)|=2$. We conclude that $G\cong S_{3}$.

$\mathbf{d_{3}=4}$

$G\cong C_{4}\text{ or }G\cong D_{8},\text{ for }d_{3}=4.$

This gives the conclusion. Moreover, we have

$\min\\{h_{m}(G)|G\text{ finite non-trivial}\\}=\frac{4}{3}\,.$

The minimum is obtained for $C_{2}$. ∎

Next we will focus on finite groups $G$ with $h_{m}(G)=3$. Note that the
smallest example of such a group is $SmallGroup(12,1)$. We are not able to
determine all these groups, but we can prove that they have even order and are
not nilpotent.

######  Proposition 2.9.

There are no finite groups $G$ of odd order with $h_{m}(G)=3$.

###### Proof.

Let $G$ be a finite group of odd order such that $h_{m}(G)=3$. Then $3||G|$.
Let $H\leq G$ with $|H|=3$. We identify the following cases:

Case 1:

$\exp(G)=3$

Then

$h_{m}(G)=\frac{|G|}{1+\frac{|G|-1}{3}}=\frac{3|G|}{|G|+2}\neq 3,$

which is a contradiction.

Case 2:

$\exp(G)\neq 3$

We will prove that

$G\text{ has at least }6\text{ elements of order }\geq 5$ (10)

We identify the cases:

  1. a)

$3^{2}|\exp(G)$

Then $G$ has at least a cyclic subgroup of order $3^{2}$ and so at least
$6=\varphi(3^{2})$ elements of order $9$.

  2. b)

$3^{2}\nmid\exp(G)$.

Let $p$ the smallest prime $\neq 3$ such that $p|\exp(G)$. If $p\geq 7$, then
$G$ has at least a cyclic subgroup of order $p$ and consequently at least
$\varphi(p)=p-1$ elements of order $p$. If $p=5$, there are two sub-cases:

    1. i)

$G$ has only one subgroup $K$ with $|K|=5$. It follows that $K\lhd G$,
therefore $KH\leq G$ and $|KH|=15$. Then $KH$ is cyclic and it possesses
$\varphi(15)=8$ elements of order $15$.

    2. ii)

$G$ has at least $3$ subgroups of order $5$. Then $G$ has at least
$3\varphi(5)=12$ elements of order $5$.

This concludes the proof of (10). We get

$h_{m}(G)\geq\frac{|G|}{1+\frac{6}{5}+\frac{|G|-7}{3}}=\frac{15|G|}{5|G|-2}>3,$

a contradiction which completes the proof.

∎

######  Proposition 2.10.

There are no finite nilpotent groups $G$ with $h_{m}(G)=3$.

###### Proof.

Assume that $G$ is a finite nilpotent group such that $h_{m}(G)=3$. If $G$ is
a $p$-group, then $p=3$ and the conclusion follows from Proposition 2.9. If
$G$ is not a $p$-group, then it can be written as a direct product of at least
two $p$-groups, say $G=G_{1}\times\cdots\times G_{k}$ with $k\geq 2$. Since

$h_{m}(G)=h_{m}(G_{1})\cdots h_{m}(G_{k}),$

we get $h_{m}(G_{i})<2$, $\forall\,i=1,...,k$. Now Theorem 2.8 implies that
$G_{i}=C_{2}^{n}$ for some $n\in\mathbb{N}$ or $G_{i}=C_{3}$, and therefore
$h_{m}(G_{i})=\frac{2^{n+1}}{2^{n}+1}$ for some $n\in\mathbb{N}$ or
$h_{m}(G_{i})=\frac{9}{5}$ . We remark that any product of these numbers is
not $3$, contradicting our assumption. ∎

Finally, we note that the results so far leave the following open question:

######  Question 2.

Which are the integer values contained in $\operatorname{Im}(h_{m})$?

## References

  * [1] M.B. Azad, B. Khosravi and H. Rashidib, On the sum of the inverses of the element orders in finite groups, Comm. Algebra 51 (2023), 694-698.
  * [2] S.J. Baishya and A. Kumar, Harmonic numbers and finite groups, Rend. Sem. Mat. Univ. Padova 132 (2014), 33-43.
  * [3] Y. Berkovich, Groups of Prime Power Order, Volume 1, de Gruyter Expositions in Mathematics 46, 2008.
  * [4] V. Grazian, C. Monetta and M. Noce, On the structure of finite groups determined by the arithmetic and geometric means of element orders, arXiv:2212.13770.
  * [5] M. Herzog, P. Longobardi and M. Maj, New criteria for solvability, nilpotency and other properties of finite groups in terms of the order elements or subgroups, Int. J. Group Theory 12 (2023), 35-44.
  * [6] I.M. Isaacs, Finite Group Theory, Amer. Math. Soc., Providence, R.I., 2008.
  * [7] M.S. Lazorec and M. Tărnăuceanu, On the average order of a finite group, J. Pure Appl. Algebra 227 (2023), article ID 107276.
  * [8] T. Leinster, Perfect numbers and groups, Eureka 55 (2001), 17–27.
  * [9] M. Tărnăuceanu and L. Tóth, Ciclicity degrees of finite groups, Acta Math. Hung. 145 (2015), 489-504.
  * [10] The GAP Group, GAP – groups, algorithms, and programming, version 4.11.0, https://www.gap-system.org, 2020.