# Decoding Fear: Exploring User Experiences in Virtual Reality Horror Games

He Zhang<EMAIL_ADDRESS>The Future Laboratory, Tsinghua UniversityNo. 160
Chengfu RoadBeijingChina College of Information Sciences and Technology,
Pennsylvania State UniversityWestgate buildingUniversity ParkUnited States
0000-0002-8169-1653 ,  Xinyang Li  Academy of Arts & Design, Tsinghua
UniversityBeijingChina 0000-0003-4081-604X ,  Christine Qiu  School of
Electrical Engineering and Computer Science, KTH Royal Institute of
TechnologyStockholmSweden 0000-0002-9069-6851 and  Xinyi Fu  The Future
Laboratory, Tsinghua UniversityNo. 160 Chengfu RoadBeijingChina
<EMAIL_ADDRESS>0000-0001-6927-0111

(2023)

###### Abstract.

This preliminary study investigated user experiences in VR horror games,
highlighting fear-triggering and gender-based differences in perception. By
utilizing a scientifically validated and specially designed questionnaire, we
successfully collected questionnaire data from 23 subjects for an early
empirical study of fear induction in a virtual reality gaming environment. The
early findings suggest that visual restrictions and ambient sound-enhanced
realism may be more effective in intensifying the fear experience.
Participants exhibited a tendency to avoid playing alone or during nighttime,
underscoring the significant psychological impact of VR horror games. The
study also revealed a distinct gender difference in fear perception, with
female participants exhibiting a higher sensitivity to fear stimuli. However,
the preference for different types of horror games was not solely dominated by
males; it varied depending on factors such as the game’s pace, its objectives,
and the nature of the fear stimulant.

Virtual reality game, affective computing, fear, coping strategy

††journalyear: 2023††copyright: rightsretained††conference: Chinese CHI 2023;
November 13–16, 2023; Denpasar, Bali, Indonesia††booktitle: Chinese CHI 2023
(CHCHI 2023), November 13–16, 2023, Denpasar, Bali, Indonesia††doi:
10.1145/3629606.3629646††isbn: 979-8-4007-1645-4/23/11††ccs: Human-centered
computing Virtual reality††ccs: Human-centered computing HCI design and
evaluation methods

##  1\. Introduction

Virtual reality (VR) has demonstrated immense potential across various fields,
particularly in human-computer interaction (HCI) (Kim, 2016). Over time,
researchers and developers have focused their attention on improving the
operability, accessibility, usability, and maintainability of VR systems.
However, as VR devices already meet a certain level of usability and
functionality, the value of enhancing the quality of VR applications based on
design thinking becomes increasingly crucial.

Emotions play an integral role in human society and have significantly
impacted the progression of human-computer interaction technologies (Picard,
1999). Particularly, they hold substantial influence in human-centered design.
Nonetheless, our grasp on human emotions, behaviors, and strategies in VR
settings is still in need of deepening (Ng and Khong, 2014).

Emotions, notably fear, are complex psychological phenomena involving
cognitive and physiological aspects (Solomon, 1980; Izard, 1991). Fear, a
strong negative emotion, is triggered by perceived or actual danger and serves
as a survival mechanism (Lynch and Martins, 2015; Clasen, 2011). The
Diagnostic and Statistical Manual of Mental Disorders (DSM-IV) (Spitzer et
al., 1994) identifies five categories of fear-provoking factors, including
animals, environment, blood/injections/injuries, situational factors, and
other factors.

VR games, particularly horror games, have proven to be effective in eliciting
fear responses from players, owing to the immersive and realistic experiences
they offer (CANTOR, 2008; Slater, 2009a; Lin, 2017; Peng et al., 2019; Lemmens
et al., 2022; Fu et al., 2021). Various game design techniques, including
architectural spaces, movement restrictions, sound effects, and jump scares,
are employed to induce fear (Kalinowski, 2019; Perron, 2004).

When fear is evoked, individuals employ various coping strategies, such as
self-help strategies, approach (monitoring) strategies, and avoidance
strategies (Lynch and Martins, 2015; Lin, 2017). Physiological responses,
including changes in heart rate, facial expressions, and specific action
tendencies, are common indicators of fear (Kivikangas, 2018).

Understanding fear stimuli is essential in studying user reflection in VR
horror environments. Fears are usually classified as innate and acquired
(Agras et al., 1969), where innate fears are inherent and include (Gibson and
Walk, 1960), for example, falling from a height, death or near death, loud
noises (Sbravatti, 2019), high-altitude flying, etc. At the same time,
experiences or memories mainly cause acquired fears, such as claustrophobia,
darkness, dentists, socializing, snakes, bugs, thunderstorms, illness,
loneliness, rejection, etc. Also, the Diagnostic and Statistical Manual of
Mental Disorders (DSM-IV) (Spitzer et al., 1994) framework classifies fear-
provoking factors into five categories: animals (e.g., dogs or spiders);
environment (e.g., fire or floods); blood/injections/injuries (e.g., wounds or
needles); situational factors (e.g., height, confined spaces or more specific
spaces such as a doctor’s office); and other factors (objectively harmless but
disturbing stimuli such as distorted faces and loud noises). These fear
factors mainly stimulate the amygdala, a region of the brain, to make people
feel fearful (LeDoux, 2012). In addition, the fear of paranormal phenomena,
such as ghosts, is often used as a stimulus to appear in games or movies
(Perron, 2018). Overall, these stimuli have three core mechanisms in the VR
environment, namely place illusion (PI, i.e., the strong illusion of being
somewhere even though one knows one is not there), plausibility illusion (PSI,
i.e., the illusion of being very real even though one inherently knows it is
not real), and body ownership (i.e., the illusion of being the real self even
though one is controlling an avatar). (i.e., although the person being
controlled is an avatar, there is still the illusion of a real self-body)
(Slater, 2009b).

##  2\. User Experiment Design

The gaming processes include the specific process of experiencing the three
games, the task settings, and the questionnaire collection.

###  2.1. VR Horror Games Selection and Usage

In this study, we selected three VR horror games known to induce fear from the
Steam platform: Richie’s Plank Experience (Toast, 2016), Phasmophobia (Games,
2020), and Emily Wants To Play (Hitchcock, 2015). The selection criteria were
as follows: a linear game script for controlling variables, game durations
between 5-20 minutes to maintain emotional arousal, simple operation rules for
easy learning, presence of elements to stimulate fear, and positive player
ratings.

Figure 1. Examples of game posters and screenshots of the actual game. The
first image on the left is the ”Richie’s Plank Experience” poster (Toast,
2016), the second image on the left is the ”Richie’s Plank Experience” in-game
screenshot, the third image on the left is the ”Phasmophobia” poster (Games,
2020), the third image on the left is the ”Phasmophobia” in-game screenshot,
the first image on the right is the ”Emily Wants To Play” in-game screenshot,
and the first image on the right is the ”Emily Wants To Play” poster
(Hitchcock, 2015).

To reduce uncontrolled variables due to individual differences, we concealed
or altered some actual goals, providing plausible but misleading objectives
(Kors et al., 2015). These allowed us to ensure the plot progression of games
2 and 3, independent of the player’s actions. For instance, in Game 2, players
were instructed to seek a nonexistent red bear toy, encouraging them to
explore dark settings and trigger more fear-inducing events. In Game 1, no
adjustments to gameplay were required as player actions did not alter the game
progression.

###  2.2. Participant

We recruited those who were interested in participating in the VR experiment
as participants through social media groups and nearby universities in China,
but for health and safety reasons, the eligible candidates for this experiment
were controlled to be between 18-30 years old and self-reported to be in good
physical health, mentally fit, and free of diseases or medical histories such
as heart disease, visual problems, and vertigo. All participants were paid a
certain amount of cash as thanks. All participants were given the option to
stay after the experimental period for a free experience with VR games for
additional time (with no special restrictions) without interfering with the
experiment or the typical research environment. This experiment was processed
under approval of university IRB. Informed consent was obtained from all
participants.

A total of 23 participants’ questionnaire data were successfully recorded,
aged between 18 and 28 years old (median age = 21) and without significant
congenital disorders. All of the participants are Chinese. All participants
completed the pretest and posttest Panas-X scales (Watson and Clark, 1994),
which is because the scale measures negative emotions and has been proven
reliable in VR environment by numerous scientific studies (Bagozzi, 1993;
Susindar et al., 2019; Drummond et al., 2021; Newman et al., 2022;
Steinhaeusser et al., 2022). Each participant who experienced a specific game
completed the questionnaire for the corresponding game.

##  3\. Human Factors Analysis based on Experiments

Gaining insights into participants’ authentic reactions to the VR horror game
experience is of paramount importance to all stakeholders. As an initial
study, our study employed both validated scientific surveys and bespoke
questionnaires. Our intent was to assemble quantifiable data from participants
about the game’s operability and its efficacy in evoking fear. This approach
furnishes a wealth of insightful and actionable data that can significantly
contribute to the trajectory of future research in this domain.

###  3.1. Analysis and Preliminary Results

We first explore the effects of VR horror games on human emotions and
reactions through scales. We set up questionnaires before and after the
experiment, and in each stage, all of the questionnaires asked participants to
base their self-perceptions on what they were feeling at the time of
completing the questionnaire. The purpose of the questionnaire was twofold:
first, to determine roughly whether the chosen VR game evokes fear through the
questionnaire statistics, and second, to explore possible future research
questions. Here, we were more descriptive in our analysis of the
questionnaire.

####  3.1.1. Game Experience Evaluation Feedback Scale

At the end of each game, we requested all participants to fill out a scale
that was designed to be called the ”Game Experience Evaluation Feedback
Scale”. This scale asked participants to rate for their feedback on the game
they had just finished experiencing. The scale consisted of ten questions,
each containing 7 rating scales (7-point Likert scale (Joshi et al., 2015)).
Participants could rate the questions on a scale from 1 to 7, where 1 is the
lowest rating (completely negative) and 7 is the highest rating (completely
positive). We first checked the reliability of the questionnaire using
Spearman-Brown reliability coefficient (Equation 1) (Eisinga et al., 2013) and
theta reliability coefficient (Equation 2) (Zumbo et al., 2007).

(1) $\rho_{xx}=2r_{hh}/(1+r_{hh})~{},$

where $\rho_{xx}$ was the reliability estimates for the entire test, $r_{hh}$
was correlation coefficient of the two halves of the test scores.

(2) $\theta=N(1-1/\lambda)/N-1~{},$

where N was the number of analysis items, $\lambda$ was root value of maximum
characteristic. The results in Table 1 and 3 showed that the value of
Spearman-Brown reliability coefficient(0.696) > 0.6 and the value of theta
reliability coefficient(0.838) > 0.8 indicated that the scale data reliability
quality was acceptable.

Reliability Statistics - Spearman-Brown coefficient |  |
---|---|---
Cronbach’s Alpha |  | Part 1 | Value | 0.237
|  |  | N of Items | 5a
|  | Part 2 | Value | 0.322
|  |  | N of Items | 5b
|  | Total N of Items |  | 10
Correlation Between Forms |  |  |  | 0.535
Spearman-Brown Coefficient |  | Equal Length |  | 0.697
|  | Unequal Length |  | 0.697
Guttman Split-Half Coefficient |  |  |  | 0.696
aThe items are: Q1, Q2, Q3, Q4, Q5. |  |  |  |
bThe items are: Q6, Q7, Q8, Q9, Q10 |  |  |  |

Table 1. Reliability Statistics - Spearman-Brown coefficient.

The specific questions and results of the scale were shown in Figure 2, where
the results included the average ratings of the participants collectively, the
results of each game, and the gender-based groupings.

Figure 2. Game Experience Evaluation Feedback Scale. The statistics were
formatted as mean (M) ± standard deviation (SD) in figure, i.e., M ± SD.
Rounding to the 2 decimal places.

The specific questions and results of the scale were shown in Figure 2, where
the results included the average ratings of the participants collectively, the
results of each game, and the gender-based groupings.

Questions 1-3 and 5 were designed to verify that the game was appropriate for
all experienced participants. Questions 4 and 6-10 were more specific to the
topic of horror games. This scale’s dataset contained 63 valid data due to the
previously mentioned problems (severe 3D vertigo and too much fear), causing
not all participants to complete all three games. The results showed that
participants generally perceived the game operation as not difficult (Q1 and
Q2 scored less than 4). Interestingly, from the results reported by the
participants, women seemed to find the operation of VR games easier than men.
At the same time, only 9(14.29%) questionnaires reported the experience of the
game as not realistic or immersive enough (Q3 scored less than 4). Analysis of
the subgroups shows that Game 2 was reported to have a more authentic
experience (5.27 $\pm$ 0.83, mean score for Q3) compared to Game 1 (4.45 $\pm$
1.10) and Game 3 (4.53 $\pm$ 0.90). Combined with the game content and the
interviews mentioned in the previous section, participants of Game 2 seem to
reflect an interesting observation that players are more likely to feel a
sense of realism in games with visual limitations (darkness), realistic
scenes, slow walking and accompanying ambient sounds (rain, footsteps, etc.).
In the Q4, participants reported feeling less pleasure or happiness during the
game experience (mean rating less than 4), which seems to be easily explained
in conjunction with Q6. Since the emotions of pleasure, happiness, and fear,
dread are opposites, inspiring fear and dread is more in line with the design
purpose of horror games. After Pearson correlation coefficient analysis, Q4
and Q6 were found to have a significant negative correlation (see Table 7). In
total, 53 reports indicated feeling very fearful and scared during the game
experience (Q6 rating greater than 4), and of these, 2 reports gave an
absolutely positive answer (rating 7). The results proved that the three games
were successful for the activation of participants’ terror emotions (overall
scores greater than 5 for Q6 and Q7). Combined with the actual data, female
participants, on average, felt more fearful than male participants in all
three games, especially in Game 3, where 100% of female participants felt a
higher level of fear rating (rating greater than or equal to 5). The results
(Q8 and Q9) showed that participants were mainly reluctant to experience the
three games in the experiment alone and would hardly choose to experience the
games at night, which may reinforce that the games have a strong activating on
the emotion of terror. Combined with the results of Q10, participants may have
lower subjective fear feelings for Game 1 relative to Game 2 and Game 3.
Females, overall, may be more sensitive to the horror elements in the game
during play.

####  3.1.2. Pre-test and Post-test PANAS-X

Emotion Scales |  |
---|---|---
Emotional Groups by Positive or Negative | Emotional Groups by Types | Emotional Items
Basic Negative Emotion Scales(1.46 $\pm$ 0.78, 1.79 $\pm$ 0.87) |  Fear(1.68 $\pm$ 1.06, 2.78 $\pm$ 1.10) |  afraid(1.96 $\pm$ 1.15, 3.04 $\pm$ 1.30), scared(1.61 $\pm$ 1.12, 3.13 $\pm$ 1.10),
|  |  frightened(1.61 $\pm$ 1.12, 3.83 $\pm$ 1.03), nervous(2.13 $\pm$ 1.29, 2.09 $\pm$ 1.12),
|  |  jittery(1.35 $\pm$ 0.65, 2.43 $\pm$ 0.84), shaky(1.67 $\pm$ 1.26, 2.17 $\pm$ 1.23)
| Hostility(1.29 $\pm$ 0.58, 1.43 $\pm$ 0.78) | angry(1.30 $\pm$ 0.56, 1.35 $\pm$ 0.65), hostile(1.30 $\pm$ 0.56, 1.61 $\pm$ 0.89),
|  | irritable(1.48 $\pm$ 0.85, 1.43 $\pm$ 0.66), scornful(1.13 $\pm$ 0.34, 1.48 $\pm$ 0.90),
|  | disgusted(1.17 $\pm$ 0.49, 1.22 $\pm$ 0.74), loathing(1.35 $\pm$ 0.65, 1.48 $\pm$ 0.85)
| Guilt(1.41 $\pm$ 0.77, 1.37 $\pm$ 0.81) | guilty(1.39 $\pm$ 0.66, 1.35 $\pm$ 0.78), ashamed(1.35 $\pm$ 0.78, 1.57 $\pm$ 1.08),
|  | blameworthy(1.39 $\pm$ 0.66, 1.30 $\pm$ 0.76), angry at self(1.39 $\pm$ 0.78, 1.39 $\pm$ 0.78),
|  | disgusted with self(1.30 $\pm$ 0.76, 1.13 $\pm$ 0.63),
|  | dissatisfied with self(1.65 $\pm$ 0.98, 1.48 $\pm$ 0.85)
| Sadness(1.45 $\pm$ 0.72, 1.56 $\pm$ 0.79) | sad(1.30 $\pm$ 0.63, 1.57 $\pm$ 0.73), blue(1.65 $\pm$ 0.83, 1.70 $\pm$ 0.82),
|  | downhearted(1.22 $\pm$ 0.42, 1.78 $\pm$ 1.04), alone(1.52 $\pm$ 0.90, 1.30 $\pm$ 0.70),
|  | lonely(1.57 $\pm$ 0.84, 1.43 $\pm$ 0.66)
Basic Positive Emotion Scales(2.54 $\pm$ 0.95, 2.33 $\pm$ 1.11) | Joviality(2.80 $\pm$ 0.96, 2.39 $\pm$ 1.29) | happy(2.78 $\pm$ 1.00, 2.26 $\pm$ 1.25), joyful(2.91 $\pm$ 0.95, 2.48 $\pm$ 1.27),
|  | delighted(2.65 $\pm$ 1.07, 2.04 $\pm$ 1.26), cheerful(2.74 $\pm$ 0.81, 2.52 $\pm$ 1.27),
|  | excited(2.78 $\pm$ 0.85, 2.74 $\pm$ 1.54), enthusiastic(2.70 $\pm$ 0.88, 2.48 $\pm$ 1.34),
|  | lively(2.83 $\pm$ 1.03, 2.13 $\pm$ 1.22), energetic(3.04 $\pm$ 1.07, 2.43 $\pm$ 1.20)
| Self-Assurance(2.34 $\pm$ 0.97, 2.00 $\pm$ 0.99) | proud(2.13 $\pm$ 0.81, 1.74 $\pm$ 0.96), strong(2.35 $\pm$ 1.07, 2.17 $\pm$ 1.03),
|  | confident(2.74 $\pm$ 0.92, 2.39 $\pm$ 1.08), bold(1.78 $\pm$ 0.85, 1.35 $\pm$ 0.65),
|  | daring(2.65 $\pm$ 1.07, 2.13 $\pm$ 1.22), fearless(2.39 $\pm$ 1.12, 2.22 $\pm$ 1.00)
| Attentiveness(2.48 $\pm$ 0.93, 2.61 $\pm$ 1.04) | alert(2.22 $\pm$ 0.95, 3.13 $\pm$ 0.76), attentive(2.91 $\pm$ 1.15, 2.65 $\pm$ 1.34),
|  | concentrating(2.78 $\pm$ 0.80, 2.78 $\pm$ 1.20), determined(2.00 $\pm$ 0.80, 1.87 $\pm$ 0.87)
Other Affective States(2.07 $\pm$ 0.9825, 2.21 $\pm$ 1.16) | Shyness(1.69 $\pm$ 0.95, 2.13 $\pm$ 1.05) | shy(1.61 $\pm$ 0.99, 2.52 $\pm$ 1.24), bashful(1.91 $\pm$ 1.04, 1.39 $\pm$ 0.66),
|  | sheepish(1.52 $\pm$ 0.85, 1.91 $\pm$ 1.04), timid(1.70 $\pm$ 1.15, 2.70 $\pm$ 1.26)
| Fatigue(1.90 $\pm$ 0.91, 2.04 $\pm$ 1.18) | sleepy(1.78 $\pm$ 0.95, 1.65 $\pm$ 1.07), tired(2.43 $\pm$ 0.95, 2.78 $\pm$ 1.35),
|  | sluggish(1.65 $\pm$ 0.83, 1.78 $\pm$ 1.13), drowsy(1.74 $\pm$ 0.92, 1.96 $\pm$ 1.15)
| Serenity(3.06 $\pm$ 1.03, 2.10 $\pm$ 1.18) | calm(3.26 $\pm$ 1.05, 2.22 $\pm$ 1.28), relaxed(3.00 $\pm$ 1.24, 2.00 $\pm$ 1.13),
|  | at ease(2.91 $\pm$ 0.79, 2.09 $\pm$ 1.12)
| Surprise(1.64 $\pm$ 1.04, 2.55 $\pm$ 1.23) | amazed(1.57 $\pm$ 0.99, 2.65 $\pm$ 1.23), surprised(1.70 $\pm$ 1.11, 2.39 $\pm$ 1.20),
|  | astonished(1.65 $\pm$ 1.03, 2.61 $\pm$ 1.27)

Table 2. Item Composition of the PANAS-X Scales and Participant Statistics.
The pre-test and post-test average scores(M) with standard deviation(SD) for
all participants are shown in parentheses after each item. Rounding to the 2
decimal places. Emotions that directly reflect fear are bolded in the table.
An example of the format is ”Item (pre-test M $\pm$ SD, post-test M $\pm$
SD)”.

At the beginning and end of the experiment, participants were asked to fill
out a PANAS-X each. The PANAS-X contains 60-item emotion descriptors with a
five-point scale system, expressing participants’ negative and positive
emotions through a higher-order scale and responding to unique emotions
through a lower-order scale (Watson and Clark, 1994). We refer to Watson and
Clark’s work (Watson and Clark, 1994) and divided the scale into three
subscales (Table 2 or Figure 3), namely (1) Basic Negative Emotion Scales,
which contains 4 emotional sets related to the emotions of fear, hostility,
guilt, and sadness respectively; (2) Basic Positive Emotion Scales, which
contains 3 emotional sets related to the emotions of joviality, self-
assurance, attentiveness and (3) Other Affective States, which contains 4
emotional sets related to the emotions of shyness, fatigue, serenity, and
surprise. Each type of emotional sets contains a number of unique emotional
items. Cronbach’s Alpha (Tavakol and Dennick, 2011) was performed on the pre-
test and post-test scales, and the results showed excellent reliability for
both the pre-test($\alpha=0.917$) and post-test($\alpha=0.919$) scales. We
conducted paired samples t-tests for the fearful emotion group on the Basic
Negative Emotion Scale, and the results(Table 5 and 6) showed significant
(p¡0.05) changes in participants’ ratings of fear-related emotions (scared,
afraid, frightened, and jittery) before the start of the experiment and after
the end of the experiment and significant (p¡0.05) correlation in
emotions(scared and afraid) between results of pre-test and post-test. As
reflected in the actual scores (Table 4), frightened emotion score increased
by 2.22, jittery emotion increased by 1.08, scared emotion increased by 1.52,
feared emotion increased by 1.08, and shaky emotion increased by 0.47. In
summary, we believe that the chosen game has a good ability to stimulate
participants’ fear. In addition to this, our descriptive statistical analysis
of the before and after results revealed some interesting phenomena, such as
that after playing the fear game, (1) participants’ sense of security
decreased substantially, while (2) they were more alert than before the game
(alert, surprise emotions went up, sleepy emotions went down), but led to
other decreases in concentration and to feelings of exhaustion (tired,
sluggish, drowsy, etc.).

The results of the user experiments showed that gender difference is a factor
in the level of fear perception and that people adopt different ways of
alleviating fear, all of which could be points of reference for future VR
horror game research.

##  4\. Discussion and Future Work

The potent induction of fear amongst participants underlines the efficacy of
VR horror games in eliciting their intended emotional responses. Intriguingly,
our findings suggest a noticeable gender difference in fear perception, with
female participants exhibiting greater sensitivity to fear stimuli. However,
preliminary results show that female players are likelier to have fun in
slower-paced games. In addition, preliminary results suggest that females do
not necessarily show more fear than males in all types of stimuli (Fig. 2. Q7
and Q10, also in Fig. 4). In investigating innate fears, (Game 1) females seem
to have a better tolerance level than males, but the results are reversed in
darkness-related environments. These revelations signal exciting opportunities
for future research to delve deeper into these gender-based discrepancies.
Exploration into physiological responses, the impact of personality traits, or
the influence of prior experiences on fear perception in VR games could
broaden our understanding of this aspect.

The preference of players for games offering visual limitations and ambient
sounds, perceived as more realistic, uncovers a promising pathway for future
VR game design. Additional research could extend upon this premise,
experimenting with diverse combinations of sensory stimuli and game mechanics
to heighten the sense of realism and immersion. An enriched understanding of
how individual elements contribute to presence can pave the way for the
development of highly immersive and thrilling VR horror games.

Our discovery that participants were reluctant to partake in the games alone
or at night intimates the profound psychological impact of these experiences.
This realization has potential implications for designing VR horror games,
particularly emphasizing user safety and well-being. Future explorations could
incorporate multiplayer modes or safety mechanisms, such as tranquil
intermissions or content filters, to modulate the intensity of fear stimuli.
Further studies could examine how these adaptations influence players’ fear
responses and overall satisfaction.

The significant surge in fear-related emotions post-experiment, coupled with a
substantial decrease in feelings of security, accentuates the profound impact
of VR horror games on users’ psychological states. As VR technology continues
to advance, becoming increasingly immersive, it becomes crucial to understand
and mitigate potential adverse effects on users’ mental health. Future
initiatives should explore strategies to provide players with post-game
support to facilitate a return to their normal emotional states. This could
include relaxation exercises or debriefing sessions.

Finally, in order to gain a more comprehensive understanding of the user’s
experience, future studies should consider incorporating semi-structured
interviews. The quantitative data collected through our questionnaire sheds
significant light on participant reactions and the elements contributing to
fear induction. However, to gain a more nuanced understanding of these
experiences, future research endeavors should consider the integration of
semi-structured interviews. This methodology can add a rich, qualitative
dimension to our data, providing detailed accounts of individual experiences,
subjective emotions, and any new observations (Qu and Dumay, 2011). The
flexibility inherent to this approach allows researchers to delve deeper into
specific questionnaire responses and further explore emergent themes or
issues. For example, such interviews can probe into why certain game elements
or environments elicit heightened fear responses in participants or why they
may be reluctant to engage with the games alone or at night. By navigating
these discussions, we can uncover the subjective insights and personal
narratives that extend beyond the reach of quantitative data.

Furthermore, semi-structured interviews could provide a platform to
investigate broader topics (Braun and Clarke, 2012), such as the potential
long-term psychological impacts of VR horror games or strategies players
employ to cope with the fear stimuli. These qualitative insights could be
instrumental in shaping future VR horror game design, ensuring they offer
immersive and thrilling experiences and consideration of user safety and
mental well-being. Although involving qualitative research might lead to a
significant increase in data volume, with the improved performance of large-
scale language models, using more automated tool (Zhang et al., 2023a) or
method (Zhang et al., 2023b) to handle such data will be promising in the
future.

Our preliminary results offer valuable insights that can guide the design of
these future semi-structured interviews. Additionally, these results could be
a meaningful reference for subsequent studies investigating fear induction in
VR horror games. By combining these approaches, we aim to paint a more
comprehensive picture of the VR horror gaming landscape and its psychological
implications.

###### Acknowledgements.

This work was supported by the National Natural Science Foundation of China
(Grant No. 62172252)

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##  Appendix A APPENDICES

Reliability Statistics |  |
---|---|---
Item |  | theta coefficient of item has been deleted | theta coefficient
Q1 |  | 0.846 | 0.838
Q2 |  | 0.846
Q3 |  | 0.848
Q4 |  | 0.797
Q5 |  | 0.804
Q6 |  | 0.827
Q7 |  | 0.824
Q8 |  | 0.798
Q9 |  | 0.803
Q10 |  | 0.812

Table 3. Reliability Statistics - theta reliability coefficient. Process in an
online statistical tool, SPSSAU. Figure 3. Item Composition of the PANAS-X
Scales and Participant Statistics. The pre-test and post-test mean scores (M)
and standard deviations (SD) of all participants are compared in this figure.
Pre-test data for the same category are lighter in color compared to post-test
data Rounding to the 2 decimal places. The original data are presented in
Table 2 Figure 4. Item Composition of the PANAS-X Scales (Fear category) and
Participant Statistics by genders. The pre-test and post-test mean scores (M)
of all participants are compared in this figure.

Paired Samples Statistics - Group of Fear Emotions
---
|  | Mean | N | Std. Deviation | Std. Error Mean
Pair 1 | frightened | 1.61 | 23 | 1.118 | .233
| post-frightened | 3.83 | 23 | 1.029 | .215
Pair 2 | jittery | 1.35 | 23 | .647 | .135
| post-jittery | 2.43 | 23 | .843 | .176
Pair 3 | scared | 1.61 | 23 | 1.118 | .233
| post-scared | 3.13 | 23 | 1.100 | .229
Pair 4 | nervous | 2.13 | 23 | 1.290 | .269
| post-nervous | 2.09 | 23 | 1.125 | .235
Pair 5 | afraid | 1.96 | 23 | 1.147 | .239
| post-afraid | 3.04 | 23 | 1.296 | .270
Pair 6 | shaky | 1.70 | 23 | 1.259 | .263
| post-shaky | 2.17 | 23 | 1.230 | .257

Table 4. Paired comparisons of each fear-related emotion(Afraid, Scared,
Frightened, Jittery, Nervous, and Shaky) before and after the experiment,
based on the results of PANAS-X scale.

Paired Differences | Significance
---|---
|  |  |  |  |  | 95% Confidence Interval
---
of the Difference
|  |  |
|  | Mean | Std. Deviation | Std. Error Mean | Lower | Upper | t | df | One-Sided p | Two-Sided p
Pair 1 | frightened - post-frightened | -2.217 | 1.413 | .295 | -2.828 | -1.606 | -7.527 | 22 | <.001* | <.001*
Pair 2 | jittery - post-jittery | -1.087 | .996 | .208 | -1.518 | -.656 | -5.234 | 22 | <.001* | <.001*
Pair 3 | scared - post-scared | -1.522 | 1.163 | .242 | -2.025 | -1.019 | -6.277 | 22 | <.001* | <.001*
Pair 4 | nervous - post-nervous | .043 | 1.581 | .330 | -.640 | .727 | .132 | 22 | .448 | .896
Pair 5 | afraid - post-afraid | -1.087 | 1.203 | .251 | -1.607 | -.567 | -4.334 | 22 | <.001* | <.001*
Pair 6 | shaky - post-shaky | -.478 | 1.442 | .301 | -1.102 | .145 | -1.591 | 22 | .063 | .126

Table 5. Paired samples t-test of fear-related emotions based on the PANAS-X.
There was a significant difference (*significant at p¡=0.05) between pre-test
and post-test in the responses to PANAS-X.

Paired Samples Correlations - Group of Fear Emotions |  |  |  |  |
---|---|---|---|---|---
|  | Significance
|  | N | Correlation | One-Sided p | Two-Sided p
Pair 1 | frightened & post-frightened | 23 | .136 | .268 | .537
Pair 2 | jittery & post-jittery | 23 | .127 | .282 | .565
Pair 3 | scared & post-scared | 23 | .450 | .016* | .031*
Pair 4 | nervous & post-nervous | 23 | .148 | .249 | .499
Pair 5 | afraid & post-afraid | 23 | .521 | .005* | .011*
Pair 6 | shaky & post-shaky | 23 | .329 | .063 | .125

Table 6. Correlation coefficients for paired samples (pre and post test) of
fear-related emotions based on the PANAS-X. There was a significant
correlation (*significant at p¡=0.05) between pre-test and post-test in the
responses to PANAS-X.

Correlations (Q4 and Q6 of Game Experience Evaluation Feedback Scale) |  |
---|---|---
|  |  | Q4 | Q6
Q4 | Pearson Correlation |  | 1 | -.288*
| Sig. (2-tailed) |  |  | .022
| N |  | 63 | 63
Q6 | Pearson Correlation |  | -.288* | 1
| Sig. (2-tailed) |  | .022 |
N | N |  | 63 | 63
* Correlation is significant at the 0.05 level (2-tailed). |  |

Table 7. Correlation Analysis of Question 4 and Question 6 of the Game
Experience Evaluation Feedback Scale.


# Finite groups which have maximal covers ††thanks: This work was supported by
NSFC (no. 11971280, 11771258).

Lifang Wang and Lijian An
Department of Mathematics, Shanxi Normal University
Linfen, Shanxi, 041004 P. R. China corresponding author. e-mail:
<EMAIL_ADDRESS>
###### Abstract

Let $\lambda(G)$ be the maximum number of subgroups in an irredundant covering
of a finite group $G$. We prove that the finite groups with
$\lambda(G)=|G|-t$, where $t\leq 5$, are solvable, and classify such groups.

Keywords covering groups, irredundant covering, solvable groups

2010 Mathematics subject classification: 20D10, 20D15

##  1 Introduction

Let $G$ be a finite group. A cover of $G$ is a collection of subgroups of $G$
whose union is $G$. The cover is irredundant if no proper sub-collection is
also a cover. Many different aspects of covers have been studied. One of these
aspects is to study the structure of groups by using the number of subgroups
in a cover of a group. Cohn [6] defined $\sigma(G)$ to be the smallest integer
$n$ such that the group $G$ is covered by $n$ proper subgroups. He proved some
properties of $\sigma(G)$ and described the structure of the groups with small
$\sigma(G)$.

Since then, many scholars study groups by $\sigma(G)$. On one hand, some
scholars calculated $\sigma(G)$ for given groups. For example, Cohn [6] proved
$\sigma(A_{5})=10$ and $\sigma(S_{5})=16$. Tomkinson [13] proved that
$\sigma(G)=1+p^{a}$, where $p^{a}$ is the order of a particular chief factor
of $G$, if $G$ is a finite solvable group. Holmes [7] determined $\sigma(G)$
for $G=M_{11},M_{22}$ and $M_{23}$ respectively. Kappe et al. [8] determined
$\sigma(S_{n})$ for $n=8,9,10,12$. On the other hand, some scholars determined
which numbers can occur as $\sigma(G)$ for groups, and, when possible, to
characterize the groups having the same value of $\sigma(G)$. For example,
Cohn [6] described the structure of the groups with $\sigma(G)=3,4,5$.
Abdollahi et al. [1] gave a description of finite groups with $\sigma(G)=6$.
Tomkinson [13] proved that there is no finite groups $G$ such that
$\sigma(G)=7$. Zhang [15] proved that there is no finite groups $G$ satisfying
$\sigma(G)=11$ or $\sigma(G)=13$. For more results, the readers refer to [2,
5, 11, 12].

As the dual of $\sigma(G)$, Rog${\rm\acute{e}}$rio [9] defined $\lambda(G)$ as
the maximum number of subgroups in an irredundant covering of a group $G$. In
[9], the author gave some basic properties of $\lambda(G)$, calculated
$\lambda(G)$ for some groups and classified the finite groups with
$\sigma(G)=\lambda(G)$ and $\lambda(G)=3,4,5$, respectively. Bastos et al. [4]
described the structure of groups $G$ with $\lambda(G)=6$.

We continue to investigate the structure of finite groups by $\lambda(G)$. In
[4, 9], the authors studied the groups with small $\lambda(G)$. In the paper,
we investigate the groups with large $\lambda(G)$ and classify the groups with
$\lambda(G)=|G|-t$, where $t\leq 5$.

The notation and terminology are standard, see [3]. We use ${\rm C}_{n},{\rm
D}_{2^{n}},{\rm Q}_{2^{n}}$ and ${\rm C}_{p}^{n}$ to denote the cyclic group
of order $n$, the dihedral group of order $2^{n}$, the generalized quaternion
group of order $2^{n}$ and the elementary abelian group of order $p^{n}$,
respectively.

Let $G$ be a finite $p$-group. For any positive integer $s$, we define

$\Omega_{s}(G)=\langle a\in G\mid a^{p^{s}}=1\rangle\ {\rm and}\
\mho_{s}(G)=\langle a^{p^{s}}\mid a\in G\rangle.$ $\Omega_{\\{s\\}}(G)=\\{a\in
G\mid a^{p^{s}}=1\\}\ {\rm and}\ \mho_{\\{s\\}}(G)=\\{a^{p^{s}}\mid a\in
G\\}.$

If $a,b$ are two elements of $G$, the commutator of $a,b$ is defined as
$[a,b]=a^{-1}b^{-1}ab$. If $H$ and $K$ are subgroups of $G$ with $G=HK$ and
$[H,K]=1$, we call $G$ a central product of $H$ and $K$, denoted by $G=H\ast
K$. Clearly, $H\cap K\leq Z(G)$.

##  2 Preliminaries

In this section, we list some results which often are used in this paper. The
following Lemma tells us $\lambda(G)$ is the number of maximal cyclic
subgroups of a finite group $G$.

######  Lemma 2.1.

[9, Proposition 4] Let $G$ be a finite group and $\langle
x_{i}\rangle,i=1,2,\cdots,n$ be the maximal cyclic subgroups of $G$. Then
$\bigcup_{i=1}^{n}\langle x_{i}\rangle$ is an irredundant covering of $G$ and
$\lambda(G)=n$.

######  Lemma 2.2.

[9, Proposition 5] Let $G$ be a finite group.
(i) $\lambda(H)\leq\lambda(G)$ for any subgroup $H$ of $G$.
(ii) If $N\unlhd G$, then $\lambda(G/N)\leq\lambda(G)$. For
$N=\cap_{i=1}^{\lambda}H_{i}$, $\lambda(G/N)=\lambda(G),$ where
$\lambda=\lambda(G)$ and $H_{1},H_{2},\cdots,H_{\lambda}$ are the maximal
cyclic subgroups of $G$.

######  Lemma 2.3.

[9, Proposition 6 and Proposition 7]
$(1)\ \lambda({\rm C}_{p}\times{\rm C}_{p^{k}})=kp-k+2;$
$(2)\ \lambda({\rm D}_{2n})=n+1;$
$(3)\ \lambda({\rm Q}_{2^{n}})=2^{n-2}+1.$

######  Lemma 2.4.

$\lambda({\rm D}_{8}\times{\rm C}_{2})=12.$

Proof Let $G={\rm D}_{8}\times{\rm C}_{2}$. Assume without loss generality

$G=\langle a,b,c\mid a^{4}=b^{2}=c^{2}=1,[a,b]=a^{2},[b,c]=[a,c]=1\rangle.$

It is clear that $\exp(G)=4$. Hence, the order of maximal cyclic subgroups is
$2$ or $4$. By Lemma 2.5, we have

$(a^{i}b^{j}c^{k})^{2}=(a^{i}b^{j})^{2}=a^{2i}[a^{i},b^{-j}]=a^{2i(1-j)}.$
$None$

Hence,

$(a^{i}b^{j}c^{k})^{2}=1\Longleftrightarrow a^{2i(1-j)}=1\Longleftrightarrow
i\equiv 0(\hbox{\rm mod }2)\ \ {\mbox{or}}\ \ j\equiv 1(\hbox{\rm mod }2).$

It follows that

$\Omega_{\\{1\\}}(G)=\\{1,b,c,bc,a^{2},a^{2}b,a^{2}c,a^{2}bc,ab,a^{3}b,abc,a^{3}bc\\}.$

Thus, the number of maximal cyclic subgroups of order $4$ is
$\frac{|G|-|\Omega_{\\{1\\}}(G)|}{2}=\frac{16-12}{2}=2$.

By $(*)$, $\mho_{\\{1\\}}(G)=\\{1,a^{2}\\}$. Hence, the number of maximal
cyclic subgroups of order $2$ is
$|\Omega_{\\{1\\}}(G)|-|\mho_{\\{1\\}}(G)|=12-2=10$. Therefore,
$\lambda(G)=10+2=12$. $\Box$

######  Lemma 2.5.

([14]) Assume $G$ is a metabelian $p$-group. Let $a,b$ be elements of $G$ and
$n$ a positive integer. Then

$(ab^{-1})^{n}=a^{n}\prod_{i+j\leq n}[ia,jb]^{n\choose{i+j}}b^{-n},$

where
$[ia,jb]=[a,b,\underbrace{a,\ldots,a}_{i-1},\underbrace{b,\ldots,b}_{j-1}].$

##  3 Some properties

By Lemma 2.1, $\lambda(G)$ is the number of maximal cyclic subgroups of $G$.
In the following, we assume that

$G=\langle g_{1}\rangle\cup\langle g_{2}\rangle\cup\cdots\cup\langle
g_{\lambda(G)}\rangle\ \ {\mbox{\rm and}}\ \ o(g_{1})\geq
o(g_{2})\geq\cdots\geq o(g_{\lambda(G)}),$

where $\langle g_{1}\rangle,\langle g_{2}\rangle,\cdots,\langle
g_{\lambda(G)}\rangle$ are all maximal cyclic subgroups of $G$.

######  Proposition 3.1.

(1) $o(g_{1})=2$ if and only if $\lambda(G)=|G|-1$.

(2) If $o(g_{1})=3$, then $\lambda(G)=|G|-k-1$, where $k$ is the number of
subgroups of order $3$ of $G$.

Proof (1) It is clear that $o(g_{1})=2$ if and only if $G\cong C_{2}^{n}$. The
result follows.

(2) Since $o(g_{1})=3$, each subgroup of order $3$ of $G$ is maximal cyclic in
$G$. Assume that $\langle g_{1}\rangle,\langle g_{2}\rangle,\cdots\langle
g_{k}\rangle$ are all subgroups of order $3$ of $G$. Let

$G=\langle g_{1}\rangle\cup\cdots\langle g_{k}\rangle\cup\langle
g_{k+1}\rangle\cdots\langle g_{\lambda(G)}\rangle,$

where $o(g_{1})=\cdots=o(g_{k})=3$. Then
$o(g_{k+1})=\cdots=o(g_{\lambda(G)})=2$. It is clear that $\langle
g_{i}\rangle\cap\langle g_{j}\rangle=1$ for $i\neq j$. Thus

$|G|=1+k(3-1)+(\lambda(G)-k)=\lambda(G)+k+1.$

It follows that $\lambda(G)=|G|-k-1$. $\Box$

By Proposition 3.1 (1), we assume that $\lambda(G)<|G|-1$ in following.

######  Proposition 3.2.

$o(g_{1})\leq|G|-\lambda(G)+1$.

Proof By Lemma 2.1, $\\{\langle g_{i}\rangle\mid i=1,\cdots,\lambda(G)\\}$ is
an irredundant covering of $G$. We have $g_{2}\cdots,g_{\lambda(G)}$ are
different pairwise elements of $G\backslash\langle g_{1}\rangle$. Hence
$\lambda(G)-1\leq|G|-o(g_{1})$. It follows that
$o(g_{1})\leq|G|-\lambda(G)+1$. $\Box$

######  Theorem 3.3.

Let $G$ be a finite non-cyclic group. If $o(g_{1})=|G|-\lambda(G)+1$, then
(1) $o(g_{2})=\cdots=o(g_{\lambda(G)})=2$.
(2) $G\cong{\rm D}_{2n}$, where $n=|G|-\lambda(G)+1$.

Proof (1) Since $G$ is non-cyclic, $\lambda(G)\geq 2$. If $o(g_{2})>2$, then,
by $o(g_{1})=|G|-\lambda(G)+1$,

$|\langle g_{1}\rangle\cup\langle
g_{2}\rangle|\geq|G|-\lambda(G)+1+2=|G|-\lambda(G)+3.$

Since $g_{3},\cdots,g_{\lambda(G)}$ are $\lambda(G)-2$ elements in
$G\backslash(\langle g_{1}\rangle\cup\langle g_{2}\rangle)$,

$\lambda(G)-2\leq|G|-|\langle g_{1}\rangle\cup\langle
g_{2}\rangle|\leq|G|-(|G|-\lambda(G)+3)=\lambda(G)-3,$

This is a contradiction. Hence $o(g_{2})=2$. Thus,
$o(g_{3})=\cdots=o(g_{\lambda(G)})=2$.

(2) Since $\lambda(G)<|G|-1$, $n>2$. It follows by (1) that $\langle
g_{1}\rangle$ is the unique cyclic subgroup of order $n$. Take any $x\in
G\backslash\langle g_{1}\rangle$. Then $o(x)=2$. Since $g_{1}x\not\in\langle
g_{1}\rangle$, $o(g_{1}x)=2$, i.e. $g_{1}xg_{1}x=1$. It follows that
$g_{1}^{x}=g_{1}^{-1}$ and $\langle g_{1},x\rangle\cong{\rm D}_{2n}$. We
assert that $G=\langle g_{1},x\rangle$. If not, then there exists an element
$y\in G\backslash\langle g_{1},x\rangle$. Since $y\not\in\langle
g_{1}\rangle$, $o(y)=2$. Thus $g_{1}^{y}=g_{1}^{-1}$. It follows that
$g_{1}^{xy}=g_{1}$. Thus $g_{1}$ and $xy$ are commute and $g_{1}xy$ is an
element of order $n$ of $G$. This is contrary to the uniqueness of $\langle
g_{1}\rangle$. Hence, $G=\langle g_{1},x\rangle\cong{\rm D}_{2n}$. $\Box$

######  Proposition 3.4.

Let $p$ be a prime. If there exists a maximal cyclic subgroup of order $p$ in
$G$, then $\lambda(G)\leq|G|-k(p-2)-1$, where $k$ is the number of maximal
cyclic subgroups of order $p$ of $G$.

Proof Let $A=\langle g_{i_{1}}\rangle\cup\langle
g_{i_{2}}\rangle\cup\cdots\cup\langle g_{i_{k}}\rangle$, where $\langle
g_{i_{1}}\rangle,\langle g_{i_{2}}\rangle,\cdots,\langle g_{i_{k}}\rangle$ are
all the maximal cyclic subgroups of order $p$ of $G$. Then $|A|=1+k(p-1)$.
Since there exist $\lambda(G)-k$ elements in $G\backslash A$,
$\lambda(G)-k\leq|G|-(1+k(p-1))$. It follows that
$\lambda(G)\leq|G|-k(p-2)-1$. $\Box$

######  Theorem 3.5.

Let $G$ be a finite group. If $\lambda(G)=|G|-t$, where $t\leq 5$, then $G$ is
solvable.

Proof By Proposition 3.2, $o(g_{1})\leq t+1$. Since $t\leq 5$, $o(g_{1})\leq
6$. If $o(g_{1})\leq 4$, then $G$ is a $\\{2,3\\}$-group. It follows by
$p^{a}q^{b}$-theorem that $G$ is solvable. If $o(g_{1})=6$, then $t=5$ and
$o(g_{2})=\cdots=o(g_{\lambda(G)})=2$ by Theorem 3.3. Hence $G$ is a
$\\{2,3\\}$-group. Hence $G$ is also solvable.

If $o(g_{1})=5$, then $t=4$ or $5$ by Proposition 3.2. We assert that $t=4$.
If not, then $t=5$ and $\lambda(G)=|G|-5$. Notice that $|\langle
g_{1}\rangle\cup\langle g_{2}\rangle|=o(g_{2})+(5-1)=o(g_{2})+4$. Since
$g_{3},\cdots,g_{\lambda(G)}\in G\backslash(\langle g_{1}\rangle\cup\langle
g_{2}\rangle)$, $\lambda(G)-2\leq|G|-|\langle g_{1}\rangle\cup\langle
g_{2}\rangle|$. It follows by $\lambda(G)=|G|-5$ that $o(g_{2})\leq 3$. If
$o(g_{2})=2$, then $o(g_{3})=\cdots=o(g_{\lambda(G)})=2$. Since $G=\langle
g_{1}\rangle\cup\langle g_{2}\rangle\cup\langle
g_{3}\rangle\cup\cdots\cup\langle g_{\lambda(G)}\rangle$,
$|G|=5+\lambda(G)-1=\lambda(G)+4$. This is contrary to $\lambda(G)=|G|-5$.
Hence, $o(g_{2})=3$. It follows that $G$ has only one cyclic subgroup $\langle
g_{1}\rangle$ of order $5$. Hence, $\langle g_{1}\rangle\unlhd G$. Clearly,
$\langle g_{1}\rangle\langle g_{2}\rangle$ is a subgroup of order $15$ of $G$.
Notice that the group of order $15$ is cyclic. However, the largest order of
maximal cyclic subgroups of $G$ is 5. This is a contradiction. Hence, $t=4$,
and hence $\lambda(G)=|G|-4$. By Proposition 3.2,
$o(g_{2})=\cdots=o(g_{\lambda(G)})=2$. Hence $|G|=2^{k}5$. It follows that $G$
is solvable. $\Box$

######  Remark 3.6.

In fact, if $t\leq 4$, then $G$ is supersolvable by Theorem 4.1, 4.2, 4.3 in
next section. However, if $t=5$, then $G$ is not necessarily supersolvable by
Theorem 4.8.

##  4 Finite groups with $\lambda(G)=|G|-t$ $(t\leq 5)$

In this section, we give a classification of finite groups with
$\lambda(G)=|G|-t(t\leq 5)$. In following, we always assume that for
$i=1,2,\cdots,{\lambda(G)}$, $g_{i}$ is as the same as the assume in the
begining of Section 3.

######  Theorem 4.1.

Let $G$ be a finite group. Then

(1) $\lambda(G)=|G|-1$ if and only if $G\cong{\rm C}_{2}^{n}$.

(2) $\lambda(G)=|G|-2$ if and only if $G\cong{\rm C}_{3}$ or $S_{3}$.

Proof (1) It is straightforward.

(2) $(\Longleftarrow).$ It is straightforward by a simple calculation.

$(\Longrightarrow).$ By Proposition 3.2, we have $o(g_{1})\leq 3$. If
$o(g_{1})=2$, then $G\cong{\rm C}_{2}^{n}$. It is clear that
$\lambda(G)=|G|-1$, which is contrary to $\lambda(G)=|G|-2$. Hence,
$o(g_{1})=3$. If $G$ is cyclic, then $G\cong{\rm C}_{3}$. Assume $G$ is non-
cyclic. Since $o(g_{1})=|G|-\lambda(G)+1$, $G\cong D_{6}\cong S_{3}$ by
Theorem 3.3. $\Box$

######  Theorem 4.2.

Let $G$ be a finite group. Then $\lambda(G)=|G|-3$ if and only if $G\cong{\rm
C}_{4}$ or ${\rm D}_{8}$.

Proof $(\Longleftarrow).$ It is straightforward by a simple calculation.

$(\Longrightarrow).$ If $G$ is cyclic, then $G\cong{\rm C}_{4}$. Assume that
$G$ is non-cyclic. By Proposition 3.2, we have $o(g_{1})\leq 4$. We assert
that $o(g_{1})=4$. If not, then $o(g_{1})=2$ or $3$. If $o(g_{1})=2$, then
$G\cong{\rm C}_{2}^{n}$. It is clear that $\lambda(G)=|G|-1$. This contradicts
$\lambda(G)=|G|-3$. If $o(g_{1})=3$, then, by Proposition 3.1,
$\lambda(G)=|G|-k-1$, where $k$ is the number of subgroups of $G$ of order
$3$. Since $\lambda(G)=|G|-3$, $k=2$. By Sylow theorem, $k\equiv 1(\hbox{\rm
mod }3)$. This contradicts $k=2$. Therefore, $o(g_{1})=4$. Since
$o(g_{1})=|G|-\lambda(G)+1$, $G\cong{\rm D}_{8}$ by Theorem 3.3. $\Box$

######  Theorem 4.3.

Let $G$ be a finite group. Then $\lambda(G)=|G|-4$ if and only if $G\cong{\rm
C}_{5}$, ${\rm D}_{10},{\rm C}_{4}\times{\rm C}_{2}$ or ${\rm D}_{8}\times{\rm
C}_{2}$.

Proof $(\Longleftarrow).$ It is clear that $\lambda({\rm C}_{5})=1$. By Lemma
2.3 and Lemma 2.4, $\lambda({\rm C}_{4}\times{\rm C}_{2})=4$, $\lambda({\rm
D}_{10})=6$ and $\lambda({\rm D}_{8}\times{\rm C}_{2})=12$. Hence
$\lambda(G)=|G|-4$.

$(\Longrightarrow).$ If $G$ is cyclic, then $G\cong{\rm C}_{5}$. Assume that
$G$ is non-cyclic. By Proposition 3.2, we have $o(g_{1})\leq 5$. We assert
$o(g_{1})=4$ or $5$. In fact, if $o(g_{1})=2$, then $G\cong{\rm C}_{2}^{n}$.
It is clear that $\lambda(G)=|G|-1$. This contradicts $\lambda(G)=|G|-4$. If
$o(g_{1})=3$. then, by Proposition 3.1, $\lambda(G)=|G|-k-1$, where $k$ is the
number of subgroups of $G$ of order $3$. Since $\lambda(G)=|G|-4$, $k=3$. By
Sylow Theorem, $k\equiv 1(\hbox{\rm mod }3)$. This contradicts $k=3$.
Therefore, $o(g_{1})=4$ or $5$.

If $o(g_{1})=5$, then $o(g_{1})=|G|-\lambda(G)+1$. Thus $G\cong D_{10}$ by
Theorem 3.3.

If $o(g_{1})=4$, then we will prove that $G\cong{\rm C}_{4}\times{\rm C}_{2}$
or ${\rm D}_{8}\times{\rm C}_{2}$ by three steps.

Step 1. $G$ is a $2$-group.

It is enough to show that $o(g_{i})\neq 3$ for $2\leq i\leq\lambda(G)$.
Suppose that there exists a maximal cyclic subgroup of order $3$ in $G$. Since
$o(g_{1})=4$, each subgroup of order $3$ is maximal cyclic in $G$. By
Proposition 3.4, $\lambda(G)\leq|G|-k-1$, where $k$ is the number of maximal
cyclic subgroups of order $3$ of $G$. Since $\lambda(G)=|G|-4$, $k\leq 3$. By
Sylow theorem, $k\equiv 1(\hbox{\rm mod }3)$. It follows that $k=1$, that is,
$G$ has unique subgroup $\langle g_{i}\rangle$ of order $3$. Hence, $\langle
g_{i}\rangle\unlhd G$.

Since $o(g_{1})=4$, $C_{G}(g_{i})=\langle g_{i}\rangle$. It follows that
$G/C_{G}(g_{i})=G/\langle g_{i}\rangle\lesssim\hbox{\rm Aut\,}(\langle
g_{i}\rangle)\cong{\rm C}_{2}$. Thus $|G|\mid 6$. This contradicts
$o(g_{1})=4$. Hence there is no subgroup of order $3$ in $G$. Thus $G$ is a
$2$-group.

Step 2. $G$ has only two maximal cyclic subgroups of order $4$ and the
intersection of them has order $2$.

Assume that $G$ has $k$ maximal cyclic subgroups $\langle
g_{1}\rangle,\cdots,\langle g_{k}\rangle$ of order $4$. Then $|\langle
g_{1}\rangle\cup\cdots\cup\langle g_{k}\rangle|\geq 2k+2$. Since
$g_{k+1},\cdots,g_{\lambda(G)}$ are different elements in $G\backslash(\langle
g_{1}\rangle\cup\cdots\cup\langle g_{k}\rangle)$,

$\lambda(G)-k\leq|G|-|\langle g_{1}\rangle\cup\cdots\cup\langle
g_{k}\rangle|\leq|G|-(2k+2).$

By $\lambda(G)=|G|-4$, we have $k\leq 2$.

We assert that $k=2$. If not, then $k=1$. By step 1, $o(g_{i})=2$ for $2\leq
i\leq\lambda(G)$. Since

$G=\langle g_{1}\rangle\cup\cdots\cup\langle g_{\lambda(G)}\rangle,$ $None$

it follows that $|G|=4+\lambda(G)-1=\lambda(G)+3$. This is contrary to
$\lambda(G)=|G|-4$. Hence, $k=2$.

Assume that $\langle g_{1}\rangle\cap\langle g_{2}\rangle=1$. Noticing that

$o(g_{1})=o(g_{2})=4,o(g_{3})=\cdots=o(g_{\lambda(G)})=2,$

we have

$|G|=4+3+\lambda(G)-2=\lambda(G)+5.$

This is a contradiction. Hence $|\langle g_{1}\rangle\cap\langle
g_{2}\rangle|=2$.

Step 3. Final results.

By Step 2, we have $\exp(G)=4$. If $G$ is abelian, then $G\cong{\rm
C}_{4}^{m}\times{\rm C}_{2}^{n}$. Since there are $6$ cyclic subgroups of
order $4$ in ${\rm C}_{4}\times{\rm C}_{4}$, $m=1$. Hence, $G\cong{\rm
C}_{4}\times{\rm C}_{2}^{n}$. Let $k$ be the number of cyclic subgroups of
order $4$ of $G$. Then

$k=\frac{|\Omega_{2}(G)|-|\Omega_{1}(G)|}{2}=\frac{2^{n+2}-2^{n+1}}{2}=2^{n}.$

It follows by step 2 that $n=1$. Thus $G\cong{\rm C}_{4}\times{\rm C}_{2}$.

Assume that $G$ is non-abelian. Notice that

$G=\langle g_{1}\rangle\cup\langle g_{2}\rangle\cup\cdots\cup\langle
g_{\lambda(G)}\rangle,$

where $o(g_{1})=o(g_{2})=4,g_{1}^{2}=g_{2}^{2}$ and
$o(g_{3})=\cdots=o(g_{\lambda(G)})=2$. We have $\mho_{1}(G)=\langle
g_{1}^{2}\rangle$. Since $G$ is a non-abelian $2$-group,
$G^{\prime}=\mho_{1}(G)=\Phi(G)=\langle g_{1}^{2}\rangle$. By [3, Lemma 4.2]
we have

$G\cong A_{1}*A_{2}*\cdots*A_{s}Z(G),\ \ {\mbox{where}}\ \ |A_{i}|=8.$

Since there exist three cyclic subgroups of order $4$ in ${\rm Q}_{8}$,
$A_{i}\ncong{\rm Q}_{8}$. It follows that $A_{i}\cong{\rm D}_{8}$.

If $s\geq 2$, then there is a subgroup $H$ is isomorphic to ${\rm D}_{8}*{\rm
D}_{8}$. Clearly, the number of cyclic subgroups of order $4$ of ${\rm
D}_{8}*{\rm D}_{8}$ is greater than $2$. This is a contradiction. Hence $s=1$.
We assert that $\exp(Z(G))=2$. If not, then there is a subgroup $H$ is
isomorphic to ${\rm D}_{8}*{\rm C}_{4}$. Clearly, the number of cyclic
subgroups of order $4$ of ${\rm D}_{8}*{\rm C}_{4}$ is greater than $2$. This
is a contradiction. Hence $\exp(Z(G))=2$. It follows that $G\cong{\rm
D}_{8}\times{\rm C}_{2}^{k}$. By step 2, $G$ has only two cyclic subgroups of
order $4$. Hence $k=1$. Thus $G\cong{\rm D}_{8}\times{\rm C}_{2}$. $\Box$

Before we classify finite groups with $\lambda(G)=|G|-5$, we need following
lemmas.

######  Lemma 4.4.

If $\lambda(G)=|G|-5$, then $o(g_{1})=3,4$ or $6$.

Proof By Proposition 3.2, $o(g_{1})\leq 6$. Since $\lambda({\rm
C}_{2}^{n})=2^{n}-1$, $o(g_{1})\neq 2$. We assert that $o(g_{1})\neq 5$. If
not, then $|\langle g_{1}\rangle\cup\langle
g_{2}\rangle|=o(g_{2})+(5-1)=o(g_{2})+4$. Since
$g_{3},\cdots,g_{\lambda(G)}\in G\backslash(\langle g_{1}\rangle\cup\langle
g_{2}\rangle)$, $\lambda(G)-2\leq|G|-|\langle g_{1}\rangle\cup\langle
g_{2}\rangle|$. It follows by $\lambda(G)=|G|-5$ that $o(g_{2})\leq 3$. If
$o(g_{2})=2$, then $o(g_{3})=\cdots=o(g_{\lambda(G)})=2$. Since $G=\langle
g_{1}\rangle\cup\langle g_{2}\rangle\cup\langle
g_{3}\rangle\cup\cdots\cup\langle g_{\lambda(G)}\rangle$,
$|G|=5+\lambda(G)-1=\lambda(G)+4$. This contradicts $\lambda(G)=|G|-5$. If
$o(g_{2})=3$, then $G$ has only one cyclic subgroup $\langle g_{1}\rangle$ of
order $5$. Hence, $\langle g_{1}\rangle\unlhd G$. Clearly, $\langle
g_{1}\rangle\langle g_{2}\rangle$ is a subgroup of order $15$ of $G$. Notice
that the group of order $15$ is cyclic. However, the largest order of maximal
cyclic subgroups of $G$ is 5. This is a contradiction. Thus $o(g_{1})=3,4$ or
$6$. $\Box$

######  Lemma 4.5.

If $\lambda(G)=|G|-5$ and $o(g_{1})=3$, then $G\cong A_{4}$, ${\rm C}_{3}^{2}$
or $\langle a,b,c\mid
a^{3}=b^{3}=c^{2}=1,a^{c}=a^{-1},b^{c}=b^{-1},a^{b}=a\rangle\cong{\rm
C}_{3}^{2}\rtimes{\rm C}_{2}$.

Proof By Proposition 3.1, there are four cyclic subgroups of order $3$. Let
$P\in\hbox{\rm Syl}_{3}(G)$ and $|P|=3^{k}$. Since $o(g_{1})=3$, $\exp(P)=3$.
Let $n_{3}$ be the number of subgroup of order $3$ of $P$. Then
$n_{3}=\frac{|P|-1}{3-1}=\frac{3^{k}-1}{2}$. Clearly $n_{3}\leq 4$. It follows
that $k\leq 2$. Thus, $P\cong{\rm C}_{3}$ or ${\rm C}_{3}^{2}$.

Case 1. $P\cong{\rm C}_{3}$.

Since $G$ has four cyclic subgroups of order $3$, $|G:N_{G}(P)|=4$ by Sylow
theorem. Let $H=N_{G}(P)$ and $H_{G}=\bigcap\limits_{x\in G}H^{x}$ be the core
of $H$. If $3\mid|H_{G}|$, then $P\leq H_{G}$. Thus, $G=N_{G}(P)H_{G}$ by
Frattini argument. Since $H=N_{G}(P)$, $H_{G}\leq N_{G}(P)$. It follows that
$G=N_{G}(P)$. Hence, $P\unlhd G$. This is a contradiction. Therefore,
$3\nmid|H_{G}|$. Since $G$ is a $\\{2,3\\}$-group and $3\nmid|H_{G}|$, $H_{G}$
is a $2$-subgroup of $G$. Noticing that $H_{G}\leq H=N_{G}(P)$, we have
$H_{G}P=H_{G}\times P$. It follows that $H_{G}=1$ by $o(g_{1})=3$. Now,
consider the transform representation of $H$. Since $|G:H|=4$ and $H_{G}=1$,
$G/H_{G}=G\lesssim S_{4}$. Noticing that $3\mid|G|$ and $4\mid|G|$, it follows
that $G\cong A_{4}$ or $S_{4}$. By $o(g_{1})=3$, we have $G\ncong S_{4}$.
Therefore, $G\cong A_{4}$.

Case 2. $P\cong{\rm C}_{3}^{2}$.

Since $o(g_{1})=3$, $G$ is a $\\{2,3\\}$-group. If $2\nmid|G|$, then
$G\cong{\rm C}_{3}^{2}$. Assume that $2\mid|G|$. By Proposition 3.1, $G$ has
only four cyclic subgroups of order $3$. It follows that $P\unlhd G$. Notice
that $o(g_{1})=3$. It follows that $G/P$ is an element abelian $2$-group.
Clearly, there is no $3^{\prime}$-elements in $C_{G}(P)$. Thus $C_{G}(P)=P$.
By the $N/C$-theorem, $G/P$ is isomorphic to an elementary abelian
$2$-subgroup of $GL(2,3)$. Let

$\alpha=\left(\begin{array}[]{cc}1&1\\\
1&0\end{array}\right),\beta=\left(\begin{array}[]{cc}1&-1\\\
0&-1\end{array}\right),$

By computation, we get $o(\alpha)=8,o(\beta)=2$ and
$\alpha^{\beta}=\alpha^{3}$. Let

$S=\langle\alpha,\beta\mid\alpha^{8}=\beta^{2}=1,\alpha^{\beta}=\alpha^{3}\rangle.$

Then $S\cong{\rm SD}_{16}.$ Hence $S$ is a Sylow $2$-subgroup of $GL(2,3)$. By
computation,
$\alpha^{4},\beta,\alpha^{2}\beta,\alpha^{4}\beta,\alpha^{6}\beta$ are all the
elements of order $2$ of $S$.

Let $P=\langle a,b\rangle$ and $1\neq c\in Q$, where $Q\in{\hbox{\rm
Syl}}_{2}(G)$. Since the largest order of elements of $G$ is 3, $c$ is an
element of order $2$ of $G$. Since $P\unlhd G$, $c$ induces an automorphism of
$P$ by conjugate. Let $a^{c}=a^{i}b^{j}$ and $b^{c}=a^{k}b^{l}$. Then the map

$\sigma:c\mapsto\left(\begin{array}[]{cc}i&j\\\ k&l\end{array}\right)$

is an isomorphic mapping from $Q$ to an elementary abelian $2$-subgroup of
$S$. Since $o(g_{1})=3$, $C_{P}(c)=1$ for any element $1\neq c\in Q$. By
computation, $\sigma(Q)=\langle\alpha^{4}\rangle$. Since
$\alpha^{4}=\left(\begin{array}[]{cc}-1&0\\\ 0&-1\end{array}\right)$,
$a^{c}=a^{-1}$ and $b^{c}=b^{-1}$. It follows that

$G=\langle a,b,c\mid
a^{3}=b^{3}=c^{2}=1,a^{c}=a^{-1},b^{c}=b^{-1},a^{b}=a\rangle.$

$\Box$

######  Lemma 4.6.

If $\lambda(G)=|G|-5$ and $o(g_{1})=4$, then $G\cong{\rm Q}_{8}$.

Proof We prove the result by the following four steps.

Step 1. $G$ is a $2$-group.

If not, then there exist maximal cyclic subgroups of order $3$ in $G$. Assume
that $G$ has $k$ maximal cyclic subgroups of order $3$. Let $A$ be the union
of the maximal cyclic subgroups of order $3$ of $G$. Then $|A|=1+2k$. Since
there are at least $\lambda(G)-(k+1)$ elements in $G\backslash(\langle
g_{1}\rangle\cup A)$, we have

$\lambda(G)-(k+1)\leq|G|-|\langle g_{1}\rangle\cup A|=|G|-(2k+4).$

It follows by $\lambda(G)=|G|-5$ that $k\leq 2$. By Sylow theorem, we have
$k=1$.

Let $H$ be the maximal cyclic subgroup of order $3$ of $G$. Then $H\unlhd G$.
Since $o(g_{1})=4$, $C_{G}(H)=H$. It follows that $G/H\lesssim\hbox{\rm
Aut\,}(H)\cong{\rm C}_{2}$. Thus $|G|=3$ or $6$. This is contrary to
$o(g_{1})=4$. Therefore, $G$ has no subgroup of order $3$. Since $o(g_{1})=4$,
$G$ is a $2$-group.

Step 2. $G$ has only three maximal cyclic subgroups of order $4$.

Let $\langle g_{1}\rangle,\langle g_{2}\rangle,\cdots,\langle g_{k}\rangle$ be
the maximal cyclic subgroups of order $4$ of $G$. Then $|\langle
g_{1}\rangle\cup\langle g_{2}\rangle\cup\cdots\cup\langle g_{k}\rangle|\geq
2k+2$. Since $g_{k+1},\cdots,g_{\lambda(G)}$ are the elements of
$G\backslash(\langle g_{1}\rangle\cup\langle g_{2}\rangle\cup\cdots\cup\langle
g_{k}\rangle)$,

$\lambda(G)-k\leq|G|-|\langle g_{1}\rangle\cup\langle
g_{2}\rangle\cup\cdots\cup\langle g_{k}\rangle|\leq|G|-(2k+2).$

It follows by $\lambda(G)=|G|-5$ that $k\leq 3$.

If $k=1$, then $o(g_{2})=\cdots=o(g_{\lambda(G)})=2$. By $G=\langle
g_{1}\rangle\cup\langle g_{2}\rangle\cup\cdots\cup\langle
g_{\lambda(G)}\rangle$, we have $|G|=4+\lambda(G)-1=\lambda(G)+3$, which is
contrary to $\lambda(G)=|G|-5$. If $k=2$, then $o(g_{1})=o(g_{2})=4$ and
$o(g_{3})=\cdots=o(g_{\lambda(G)})=2$. By $G=\langle g_{1}\rangle\cup\langle
g_{2}\rangle\cup\cdots\cup\langle g_{\lambda(G)}\rangle$, we have
$|G|=|\langle g_{1}\rangle\cup\langle g_{2}\rangle|+\lambda(G)-2$. Since
$\lambda(G)=|G|-5$, $|\langle g_{1}\rangle\cup\langle g_{2}\rangle|=7$. It
follows that $\langle g_{1}\rangle\cap\langle g_{2}\rangle=1$. By step 1, $G$
is a $2$-group. Hence $Z(G)\neq 1$. Take an element $a$ of order $2$ in
$Z(G)$. Then $a\not\in\langle g_{1}\rangle$ or $a\not\in\langle g_{2}\rangle$.
Without losing generality, assume that $a\not\in\langle g_{1}\rangle$. Then
$\langle ag_{1}\rangle$ is a cyclic subgroup of order $4$. Since $k=2$ and
$a\not\in\langle g_{1}\rangle$, $\langle ag_{1}\rangle=\langle g_{2}\rangle$.
Thus $g_{2}^{2}=(ag_{1})^{2}=g_{1}^{2}\in\langle g_{1}\rangle\cap\langle
g_{2}\rangle$, which is contrary to $\langle g_{1}\rangle\cap\langle
g_{2}\rangle=1$. To sum up, $k=3$. That is, $G$ has only three maximal cyclic
subgroups $\langle g_{1}\rangle,\langle g_{2}\rangle$ and $\langle
g_{3}\rangle$ of order $4$.

Step 3. $\langle g_{1},g_{2},g_{3}\rangle\cong Q_{8}$.

If $|\langle g_{1}\rangle\cap\langle g_{2}\rangle\cap\langle g_{3}\rangle|=1$,
then $|\langle g_{1}\rangle\cup\langle g_{2}\rangle\cup\langle
g_{3}\rangle|\geq 9$. Noticing that $G=\langle g_{1}\rangle\cup\langle
g_{2}\rangle\cup\cdots\cup\langle g_{\lambda(G)}\rangle$ and
$o(g_{4})=\cdots=o(g_{\lambda(G)})=2$, we have

$|G|=|\langle g_{1}\rangle\cup\langle g_{2}\rangle\cup\langle
g_{3}\rangle|+\lambda(G)-3\geq 9+\lambda(G)-3=\lambda(G)+6.$

It follows that $\lambda(G)\leq|G|-6$, which is contrary to
$\lambda(G)=|G|-5$. Hence $|\langle g_{1}\rangle\cap\langle
g_{2}\rangle\cap\langle g_{3}\rangle|\neq 1$ and
$g_{1}^{2}=g_{2}^{2}=g_{3}^{2}$.

We claim that $\langle g_{1},g_{2},g_{3}\rangle$ is not abelian. Otherwise,
$\langle g_{1}g_{2}g_{3}\rangle$ is the 4th maximal cyclic subgroup of order
4, a contradiction. Without loss of generality, we assume that
$[g_{1},g_{2}]\neq 1$. It is easy to see that $\langle g_{1},g_{2}\rangle\cong
Q_{8}$ and $g_{3}=g_{1}g_{2}$.

Step 4. $G\cong Q_{8}$.

By

$G=\langle g_{1}\rangle\cup\langle g_{2}\rangle\cup\cdots\cup\langle
g_{\lambda(G)}\rangle$

and

$o(g_{1})=o(g_{2})=o(g_{3})=4,o(g_{4})=\cdots=o(g_{\lambda(G)})=2,g_{1}^{2}=g_{2}^{2}=g_{3}^{2},$

we have $\Phi(G)=\mho_{1}(G)=\langle g_{1}^{2}\rangle$. It follows that
$\langle g_{1}\rangle\unlhd G$.

We assert $C_{G}(g_{1})=\langle g_{1}\rangle$. If not, there exists an element
$a\in C_{G}(g_{1})\backslash\langle g_{1}\rangle$. It is easy to see that
$\langle g_{1}a\rangle$ is the 4th maximal cyclic subgroup of order 4, a
contradiction. Hence $C_{G}(g_{1})=\langle g_{1}\rangle$. By $N/C$-theorem,
$G/C_{G}(g_{1})=G/\langle g_{1}\rangle\lesssim C_{2}$. Thus, $|G|\leq 8$. By
Step 3, $G=\langle g_{1},g_{2},g_{3}\rangle\cong{\rm Q}_{8}$. $\Box$

######  Lemma 4.7.

If $\lambda(G)=|G|-5$ and $o(g_{1})=6$, then $G\cong{\rm C}_{6}$ or ${\rm
D}_{12}$.

Proof If $G$ is cyclic, then $\lambda(G)=1$. Since $\lambda(G)=|G|-5$, we get
$G\cong{\rm C}_{6}$. Assume that $G$ is non-cyclic. Since
$o(g_{1})=|G|-\lambda(G)+1$, $G\cong{\rm D}_{12}$ by Theorem 3.3. $\Box$

######  Theorem 4.8.

Let $G$ be a finite group. Then $\lambda(G)=|G|-5$ if and only if $G$ is
isomorphic to one of the groups in Lemma 4.5, 4.6 and 4.7.

Proof $(\Longrightarrow).$ It follows by Lemma 4.4–4.7.

$(\Longleftarrow).$ It is clear that $\lambda({\rm C}_{6})=1$. By Lemma 2.3,
$\lambda({\rm Q}_{8})=3$, $\lambda({\rm D}_{12})=7$ and $\lambda({\rm
C}_{3}^{2})=4$. It is clear that the order of maximal cyclic subgroup of
$A_{4}$ is $2$ or $3$. Since there are three cyclic subgroups of order $2$ and
four cyclic subgroups of order $3$, $\lambda(A_{4})=7$.

Assume that $G=\langle a,b,c\mid
a^{3}=b^{3}=c^{2}=1,a^{c}=a^{-1},b^{c}=b^{-1},a^{b}=a\rangle$. Let $P=\langle
a,b\rangle$. Then $P\in\hbox{\rm Syl}_{3}(G)$ and $P\unlhd G$. It is clear
that $x^{c}=x^{-1},\forall x\in P$. Thus $N_{G}(\langle c\rangle)=\langle
c\rangle$ and the order of maximal cyclic subgroups of $G$ is $2$ or $3$. It
follows that $G$ has four cyclic subgroups of order $3$ and nine cyclic
subgroups of order $2$. Thus $\lambda(G)=13$.

To sum up, $\lambda(G)=|G|-5$ for the groups listed in Lemma 4.5, 4.6 and 4.7.
$\Box$

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