# Characterizing some finite groups by the average order

Ashkan ZareZadeh1 ,  Behrooz Khosravi1 and  Zeinab Akhlaghi1,2 1 Faculty of
Mathematics and Computer Science, Amirkabir University of Technology (Tehran
Polytechnic), 15914 Tehran, Iran.  2 School of Mathematics, Institute for
Research in Fundamental Science (IPM) P.O. Box:19395-5746, Tehran, Iran.
(A. Zarezadeh<EMAIL_ADDRESS>
(B. Khosravi<EMAIL_ADDRESS>
(Z. Akhlaghi<EMAIL_ADDRESS>
###### Abstract.

The average order of a finite group $G$ is denoted by $\mathrm{o}(G)$. In this
note, we classify groups whose average orders are less than
$\mathrm{o}(S_{4})$, where $S_{4}$ is the symmetric group on four elements.
Moreover, we prove that $G\cong S_{4}$ if and only if
$\mathrm{o}(G)=\mathrm{o}(S_{4})$. As a consequence of our results we give a
characterization for some finite groups by the average order. In [9, Theorem
1.2], the groups whose average orders are less than $\mathrm{o}(A_{4})$ are
classified. It is worth mentioning that to get our results we avoid using the
main theorems of [9] and our results leads to reprove those theorems.

###### Key words and phrases:

Element order, sum of element orders, average order, characterization

###### 2000 Mathematics Subject Classification:

05C25, 05C69, 94B25

The first author is supported by a grant from IPM (No. 1402200112)

##  1\. Introduction

For a finite group $G$, $\psi(G)$ was first introduced in [1], which denotes
the sum of the element orders of $G$, i.e., $\psi(G)=\sum\limits_{x\in
G}{o(x)}$. Later the average order of $G$ was defined as,
$\mathrm{o}(G)=\frac{\psi(G)}{|G|}$.

At first glance, these quantities are not expected to inform us much about the
structure of $G$. As an example, $\psi(A_{4})=\psi(D_{10})=31$, and also
$\mathrm{o}(D_{12})=\mathrm{o}(C_{4})=2.75$. If $G$ is a group such that there
exist(s) exactly $k$ non-isomorphic groups with average order $\mathrm{o}(G)$,
then we say $G$ is k-recognizable by average order. A $1$-recognizable group
is called characterizable. In [5], it is conjectured that
$\mathrm{o}(G)<\mathrm{o}(A_{5})$, guaranties the solvability of $G$, which
turned out to be true when Herzog, Longobardi and Maj proved it in [3]. In
that paper, they also classified all finite groups with
$\mathrm{o}(G)\leq\mathrm{o}(S_{3})$. Later in [4], they proved that $A_{5}$
is characterizable by average order, meaning that
$\mathrm{o}(G)=\mathrm{o}(A_{5})$ implies $G\cong A_{5}$. Meanwhile
Tărnăuceanu in [9], classified all finite groups with
$\mathrm{o}(G)\leq\mathrm{o}(A_{4})$, where he stated that if
$\mathrm{o}(G)<\mathrm{o}(A_{4})$, $G$ would be supersolvable, and
$\mathrm{o}(G)=\mathrm{o}(A_{4})$ leads to $G\cong A_{4}$. As the main result
of this paper, inspired by the above results, we determine the groups whose
average orders are less than $2.8$. Note that the bound $2.8$ is close to
$\mathrm{o}(S_{4})=\frac{67}{24}\approx 2.791$. As a consequence, we give a
new characterization for some finite groups using the average order. As the
main result, we prove the following theorem:

######  Main Theorem.

Let $G$ be a finite group satisfying $\mathrm{o}(G)<2.8$. Then $G$ is
isomorphic to one of the following groups:

  1. it(i)

$D_{12}$;

  2. it(ii)

$S_{4}$;

  3. it(iii)

$C_{3}$ or $C_{3}\times C_{3}$;

  4. it(iv)

$C_{2}^{2k}\rtimes C_{3}$, for some natural number $k$, which is a Frobenius
group;

  5. it(v)

a $2$-group of one of the following types, where $E\cong C_{2}^{m}$, for some
$m\geq 0$:

    1. it(1)

$C_{4}$;

    2. it(2)

$D_{8}\times E$;

    3. it(3)

an elementary abelian $2$-group;

    4. it(4)

$D(C_{4}\times C_{4})\times E$, where $D(C_{4}\times C_{4})$ is the
generalized dihedral of $C_{4}\times C_{4}$;

    5. it(5)

$(D_{8}*D_{8})\times E$, where $(D_{8}*D_{8})$ is the central product of
$D_{8}$ and $D_{8}$;

    6. it(6)

$S(2)\times E$, where $S(2)=\langle
c,x_{1},y_{1},x_{2},y_{2}|c^{2}=x_{i}^{2}=y_{i}^{2}=e$, all pairs of
generators commute except $[c,x_{i}]=y_{i}\rangle$.

  6. it(vi)

$C_{3}^{k}\rtimes C_{2}$, for some natural number $k$, which is a Frobenius
group.

Determining finite groups whose average orders are less than $2.8$ can also be
helpful in studying the density of the quantity $\mathrm{o}(G)$ which was
discussed in [6]. To prove our results, we do not use [9, Theorem 1.2] and as
a consequence, we pose another proof for those results.

Throughout this paper, all groups are non-trivial finite groups, for
$\theta\in{\mathrm{Aut}}(G)$, $r(G,\theta)$ denotes $n(\theta)/|G|$, where
$n(\theta)$ is the number of the elements mapped to their inverses by
$\theta$, $n_{k}(A)$ denotes the number of the elements of order $k$ in
$A\subseteq G$, and $\pi(G)$ is the set of all prime divisors of $|G|$, $H(r)$
is the central product of $r$ copies of $D_{8}$, $S(r)=\langle
c,x_{1},y_{1},x_{2},y_{2},...,x_{r},y_{r}|c^{2}=x_{i}^{2}=y_{i}^{2}=e$, all
pairs of generators commute except $[c,x_{i}]=y_{i}\rangle$, and $D(A)$ is the
generalized dihedral group of abelian group $A$.

######  Corollary.

By the notations in the main theorem:

  1. it(a)

if $G$ is an elementary abelian $2$-group, or isomorphic to $D_{8}\times
C_{2}^{k}$, for some integer $k\geq 0$, or one of the groups mentioned in
Cases $(ii)$, $(iii)$, $(iv)$, $(vi)$, then $G$ is characterizable by average
order,

  2. it(b)

if $G$ is isomorphic to $D_{12}$ or $C_{4}$, then $G$ is $2$-recognizable by
average order,

  3. it(c)

if $G$ is isomorphic to the groups stated in (4), (5) or (6) of Case $(v)$,
then $G$ is $3$-recognizable by average order.

##  2\. preliminary results

The following lemmas are used several times in the proof of our main results.
The first lemma is applied in the proofs multiple times without any further
reference.

From the proof of [3, Lemma 3.1], we have the following lemma:

######  Lemma 2.1.

Let $G$ be a group, and $H$ a non-trivial normal subgroup of $G$.

  1. (i)

For each $x\in G$ and $h\in H$, $o(xH)\mid o(xh)$,

  2. (ii)

$\frac{\psi(H)-|H|}{|G|}+\mathrm{o}(\frac{G}{H})\leq\mathrm{o}(G)$. In
particular, $\mathrm{o}(\frac{G}{H})<\mathrm{o}(G)$.

######  Lemma 2.2.

[7, Theoren 2.5] Let $\theta$ be an automorphism of $G$ and $P$ a Sylow
$p$-subgroup of $G$. If $r(G,\theta)>\frac{2}{p+1}$, then $P$ is normal in
$G$.

######  Lemma 2.3.

Let $K$ be a minimal normal subgroup of a finite group $G$. If $|K|$ is odd
and $G/K\cong C_{2}\times C_{2}$, then $|K|=p$, for some prime $p$ and $G\cong
C_{2}\times C_{2p}$ or $G\cong D(C_{2p})$.

###### Proof.

Note that $G$ is solvable, so $K$ is an elementary abelian $p$-subgroup, for
some odd prime $p$. Since $G$ is not a Frobenius group, there exists a
subgroup $M$ of $G$ of index $2$, where $M$ is not a Frobenius group. Let $Q$
be a Sylow 2-subgroup of $M$. As $C_{K}(Q)\leq K$ is the normal Sylow
$p$-subgroup of ${\bf Z}(M)$, we get that $C_{K}(Q)=K$, and so $G\cong
C_{2}\times(K\rtimes C_{2})$ or $G\cong C_{2}\times C_{2}\times K$. In both
cases, since $K$ is minimal normal, $|K|=p$, as wanted.

######  Lemma 2.4.

Let $\frac{m}{n}$ be a rational number, where $(m,n)=1$, and $m$ is even.
There exists no finite group $G$ such that $\mathrm{o}(G)=\frac{m}{n}$.

###### Proof.

Assume that there exists a finite group $G$ such that
$\mathrm{o}(G)=\frac{m}{n}$. Then by the definition of $\mathrm{o}(G)$,
$\psi(G)=\frac{m}{n}|G|$. Now by the fact that $\psi(G)$ is always odd, we get
a contradiction.

Therefore, $\mathrm{o}(G)$ can not be equal to $2.4$, $2.8$, $3.6$, or any
even integer, for any finite group $G$.

######  Lemma 2.5.

Let $G$ be a finite nilpotent group. Then the following statements hold.

  1. (i)

If $|\pi(G)|>1$, then $\mathrm{o}(G)\geq\mathrm{o}(C_{6})=3.5$, and if
$G\ncong C_{6}$, then $\mathrm{o}(G)>4$.

  2. (ii)

If $|G|>5$ is odd and $G$ is not a $3$-group, then
$\mathrm{o}(G)\geq\mathrm{o}(C_{5}^{2})=4.84$.

###### Proof.

($i$) By [3, Lemma 1.1], if $\pi(G)=\\{p_{1},p_{2},...,p_{n}\\}$, then
$\mathrm{o}(G)=\prod\limits_{1\leq i\leq n}{o({{P}_{i}})}$, where
$P_{i}\in\mathrm{Syl}_{p_{i}}(G)$. If $|G|$ has a prime divisor $p\geq 5$,
then $G$ has a quotient isomorphic to its Sylow $p$-subgroup, say P, and since
$\mathrm{o}(P)\geq\mathrm{o}(C_{p})>4$, the statement holds. Hence, we may
assume that $\pi(G)=\left\\{2,3\right\\}$. If $G\ncong C_{6}$, then
$12\mid|G|$ or $18\mid|G|$. Therefore,
$\mathrm{o}(G)\geq\mathrm{o}(C_{2}^{2})\mathrm{o}(C_{3})>4$.

($ii$) If $|G|$ has a prime divisor $p\geq 7$, then $G$ has a quotient
isomorphic to its Sylow $p$-subgroup, say $P$, and since
$\mathrm{o}(P)\geq\mathrm{o}(C_{p})>6>\mathrm{o}(C_{5}^{2})$, the statement
holds. Otherwise, $\pi(G)=\left\\{5\right\\}$, or
$\pi(G)=\left\\{3,5\right\\}$. In the first case, $G$ is a $5$-group and as
$|G|>5$, the statement holds. In the second case, since $G$ has a factor,
isomorphic to $C_{15}$, we have
$\mathrm{o}(G)\geq\mathrm{o}(C_{15})=9.8>\mathrm{o}(C_{5}^{2})$.

######  Lemma 2.6.

Let $G$ be a finite group and $H$ a subgroup of $G$.

  1. (i)

If $H$ is of index $2$, and $\mathrm{o}(H)>3.6$, then $\mathrm{o}(G)>2.8$.

  2. (ii)

If $H$ is a normal subgroup of index $3$ and $\mathrm{o}(H)>2.4$, then
$\mathrm{o}(G)>2.8$.

###### Proof.

($i$) As $G=H\cup xH$, for some $x\in G\setminus H$, we have,
$\mathrm{o}(G)\geq\frac{\psi(H)+2|H|}{2|H|}=\frac{\mathrm{o}(H)}{2}+1>2.8$.
The proof of ($ii$) is similar to ($i$). By Lemma 2.1,
$\mathrm{o}(G)\geq\frac{\psi(H)+6|H|}{3|H|}=\frac{\mathrm{o}(H)}{3}+2>2.8$, as
wanted.

######  Lemma 2.7.

Let $M$ be a subgroup of index $2$ of a finite group $G$. If $n_{2}(G\setminus
M)>2|M|/3=|G|/3$, then $M$ is nilpotent.

###### Proof.

Let $x\in G\setminus M$ be an involution and $\tau_{x}\in{\mathrm{Aut}}(M)$,
such that for each $m\in M$, $\tau_{x}(m)=m^{x}$. Note that $\tau_{x}$ maps
$m\in M$ to its inverse if and only if $o(xm)=2$. Since $n_{2}(G\setminus
M)>2|M|/3$, we get that $\tau_{x}$ maps more than $\frac{2}{3}$ of the
elements of $M$ to their inverses. As,
$r(M,\tau_{x})>\frac{2}{3}\geq\frac{2}{p+1}$, for any prime number $p$, by
Lemma 2.2, $M$ is nilpotent.

##  3\. main results

######  Lemma 3.1.

Let $G$ be a finite group and $m>1$ be an odd integer.

  1. (i)

If $|G|=2m$, then $\mathrm{o}(G)\geq\mathrm{o}(D(C_{5}\times C_{5}))=3.42$,
unless $G\cong D(C_{3}^{k})$, for some integer $k$, or $G\cong D_{10}$.

  2. (ii)

If $|G|=4m$, then $\mathrm{o}(G)>3$, unless $G\cong A_{4}$ or $D_{12}$.

###### Proof.

($i$) Let $N$ be the Hall $2^{\prime}$-subgroup of $G$, and $G=N\cup xN$, for
some involution $x\in G\setminus N$. Note that $N$ is a group of odd order, so
by [3, Lemma 1.1], $\mathrm{o}(N)\geq\mathrm{o}(C_{3})=\frac{7}{3}$. Note that
for any $n\in N$, $o(xn)=2$ if and only if $n^{x}=n^{-1}$. Hence, by [2, Lemma
10.4.1], $n_{2}(xN)=n_{2}(G\setminus N)$ is a divisor of $|N|$. If
$n_{2}(G\setminus N)\leq\frac{|N|}{3}=\frac{|G|}{6}$, then $\psi(xN)\geq
2n_{2}(xN)+6(|N|-n_{2}(xN))\geq 2\frac{|G|}{6}+6\frac{|G|}{3}=\frac{7}{3}|G|$.
Thus,
$\mathrm{o}(G)\geq\frac{\mathrm{o}{(N)}}{2}+\frac{7}{3}\geq\frac{7}{6}+\frac{7}{3}=3.5$.
If $n_{2}(G\setminus N)>\frac{|N|}{3}$, we see that $n_{2}(xN)=|N|$. So
$\langle x\rangle$ acts Frobeniusly on $N$. Therefore, $N$ is abelian. First
assume that $N$ is a 3-group. If exp($N$)$>3$, then $N$ has a quotient
isomorphic to $C_{9}$, and so $\mathrm{o}(N)\geq\mathrm{o}(C_{9})>6$. Hence,
$\mathrm{o}(G)=\frac{\mathrm{o}(N)}{2}+1>3+1=4$. Otherwise, exp($N$)$=3$, and
so $G\cong D(C_{3}^{k})$, for some $k$, as wanted. If $N$ is not a $3$-group
and $m>5$, then by Lemma 2.5($ii$), $\mathrm{o}(N)\geq\mathrm{o}(C_{5}\times
C_{5})=4.84$, and so $\mathrm{o}(G)=\frac{\mathrm{o}(N)}{2}+1\geq
2.42+1=3.42$. Finally, for $m=5$, $G\cong D_{10}$, and
$\mathrm{o}(D_{10})=3.1$, as desired.

($ii$) Let $Q\in$ Syl${}_{2}(G)$. If $Q$ is cyclic, then $G\cong N\rtimes Q$,
where $N$ is the Hall $2^{\prime}$-subgroup of $G$. Since $N$ is of odd order,
$\mathrm{o}(N)\geq\mathrm{o}(C_{3})=\frac{7}{3}$. By Lemma 2.1,
$\mathrm{o}(G)\geq\frac{\mathrm{o}(N)}{4}+\frac{10}{4}\geq\frac{7}{12}+\frac{10}{4}=\frac{37}{12}>3$.
Now, assume that $Q$ is not cyclic. Let $m=3$. For the nilpotent groups of
order 12, by Lemma 2.5, $\mathrm{o}(G)>3$. Looking at the non-nilpotent groups
of order $12$ with non-cyclic Sylow $2$-subgroups, we see that the only
possibilities are $D_{12}$ and $A_{4}$. From now on, we assume that $m\geq 5$,
and by induction on $m$, we prove that $\mathrm{o}(G)>3$.

Note that by [3, Theorem C], if $G$ is non-solvable, then
$\mathrm{o}(G)>\mathrm{o}(A_{5})>3.5$, and the result holds. If $G$ is
solvable, there exists a maximal normal subgroup $M$ of index $p$, for some
prime $p$. From the fact that
$p-1+\frac{1}{p}=\mathrm{o}(C_{p})=\mathrm{o}(\frac{G}{M})<\mathrm{o}(G)$, if
$p\geq 5$, we get that $\mathrm{o}(G)>3$. In the sequel, we consider
$p\in\left\\{2,3\right\\}$. Checking by GAP, we get that the statement holds
for groups of order $36$ and so in the following we may assume that $|G|\neq
36$. Now we consider the following cases.

Case 1) If $p=2$, then $G$ has a normal $2$-complement, say $K$. Let
$K_{0}\leq K$ be a minimal normal subgroup of $G$. If $K_{0}<K$, we have
$|G:K_{0}|=4m^{\prime}$, where $m^{\prime}\geq 3$. If $m^{\prime}\geq 5$, by
the induction hypothesis, $\mathrm{o}(G)>\mathrm{o}(G/K_{0})>3$. Let
$m^{\prime}=3$. So, $|G:K_{0}|=12$. Note that since $A_{4}$ does not have a
normal $2$-complement, $G/K_{0}\ncong A_{4}$. If $G/K_{0}\ncong D_{12}$, then
by the discussion we had for groups of order $12$,
$\mathrm{o}(G)>\mathrm{o}(G/K_{0})>3$. Otherwise, $G/K_{0}\cong D_{12}$. Let
$M$ be a subgroup of $G$ such that $M/K_{0}\cong C_{6}$. So,
$\mathrm{o}(M)>\mathrm{o}(C_{6})=3.5$. If $n_{2}(G\setminus M)\leq|G|/3$, then
$\mathrm{o}(G)\geq\frac{\psi(M)+2|G|/3+6|G|/6}{|G|}\geq\frac{7}{4}+\frac{5}{3}>3$.
Otherwise, $n_{2}(G\setminus M)>|G|/3$, then by Lemma 2.7, $M$ is nilpotent.
Thus, by Lemma 2.5($i$), $\mathrm{o}(M)>4$. Therefore,
$\mathrm{o}(G)\geq\frac{\mathrm{o}(M)}{2}+1>3$, as desired. Now assume that
$K_{0}=K$. Therefore, by Lemma 2.3, $G\cong D(C_{2q})$ or $G\cong C_{2q}\times
C_{2}$, where $q>3$. In the first case,
$\mathrm{o}(G)=\frac{\mathrm{o}(C_{q}\times C_{2})}{2}+1>3$. In the latter
case, by Lemma 2.5($i$), $\mathrm{o}(G)>3$.

Case 2) If $p=3$, then $G=M\cup xM\cup x^{2}M$, for some $x\in G\setminus M$.
On the other hand, by the induction hypothesis, $\mathrm{o}(M)>3$. So,
$\mathrm{o}(G)\geq\frac{\mathrm{o}(M)}{3}+2>3$.

######  Lemma 3.2.

Let $G$ be a finite group with $|\pi(G)|\geq 2$. Then
$\mathrm{o}(G)\geq\mathrm{o}(S_{3})=\frac{13}{6}$, and equality holds if and
only if $G\cong S_{3}$.

###### Proof.

For the groups of order $6$, the statement holds. So we may assume that
$|G|\geq 10$. If $G$ is of odd order, as exp($G$)$\geq 3$, $\mathrm{o}(G)\geq
3-\frac{2}{|G|}>2.86>\frac{13}{6}$. Now assume that $G$ is of even order. If
$n_{2}(G)>\frac{2}{3}|G|-1$, then since $r(G,1_{G})>\frac{2}{3}$, by Lemma
2.2, the Sylow $2$-subgroup of $G$, say $P$, is normal in $G$. Now because
$|G:P|<\frac{|G|}{n_{2}(G)+1}<\frac{3}{2}$, $G$ is a $2$-group, a
contradiction. Hence, $n_{2}(G)\leq\frac{2}{3}|G|-1$. As, $\psi(G)\geq
1+2n_{2}(G)+3(|G|-n_{2}(G)-1)\geq\frac{7}{3}|G|-1$, we have
$\mathrm{o}(G)\geq\frac{7}{3}-\frac{1}{|G|}>\mathrm{o}(S_{3})$, since $|G|\geq
10$.

Previous lemma implies that $\mathrm{o}(G)\leq S_{3}$, leads to $G\cong
S_{3}$, or $G$ is a $2$-group. Now by [10, Corollary], we get that $G$ is an
elementary abelian $2$-group, if $G\ncong S_{3}$, which is a new proof for [3,
Theroem A].

The following lemma is obtained by [9, Theorem 1.2], but we prove it without
referring to that result.

######  Lemma 3.3.

If $\mathrm{o}(G)<2.4$, then $G$ is isomorphic to one of the following groups:

  1. (i)

$C_{3}$;

  2. (ii)

$D_{8}$;

  3. (iii)

$S_{3}$;

  4. (iv)

$D(C_{3}\times C_{3})$;

  5. (v)

an elementary abelian $2$-group.

###### Proof.

We note that the result holds when $|G|\leq 10$. So we may assume that
$|G|\geq 11$. If $|G|$ is odd, as $exp(G)\geq 3$, it is easy to check that
$\mathrm{o}(G)\geq 3-2/|G|>\mathrm{o}(C_{3}\times C_{3})>2.7$, a
contradiction. Now assume that $|G|$ is even. Obviously the statement holds
for elementary abelian $2$-groups and if $G$ is a $2$-group which is not
elementary abelian, then by [10, Corollary], $n_{2}(G)\leq\frac{3}{4}|G|-1$.
Therefore,
$\mathrm{o}(G)\geq\frac{1+2n_{2}(G)+4(|G|-n_{2}(G)-1)}{|G|}=2.5-\frac{1}{|G|}$,
and since $\mathrm{o}(G)<2.4$, we have $|G|\leq 8$, a contradiction. In the
sequel, suppose that $G$ is not a $2$-group. If there exists a non-normal
Sylow $p$-subgroup $P$ of $G$, for some odd prime $p$, then by considering
$1_{G}\in{\mathrm{Aut}}(G)$ in Lemma 2.2, we have
$r(G,1_{G})\leq\frac{2}{p+1}$. It follows that
$n_{2}(G)\leq\frac{2}{p+1}|G|-1$. So, $\psi(G)\geq
1+2n_{2}(G)+3(|G|-n_{2}(G)-1)>(3-\frac{2}{p+1})|G|-1$. Therefore,
$\mathrm{o}(G)\geq 2.5-\frac{1}{|G|}>2.4$, since $p\geq 3$ and $|G|\geq 11$, a
contradiction. Therefore, $G\cong H\rtimes Q$, where $H$ is the Hall
$2^{\prime}$-subgroup of $G$, and $Q$ is a Sylow $2$-subgroup of $G$. Note
that as we discussed in Lemma 3.1, if $|Q|\leq 4$, then $G\cong D(C_{3}\times
C_{3})$, as desired. Let $|Q|\geq 8$ and $M$ be a subgroup of index 2 of $G$.
If $n_{2}(G\setminus M)\leq\frac{|G|}{3}$, as it was discussed multiple times,
$\mathrm{o}(G)\geq\frac{\mathrm{o}(M)}{2}+\frac{4}{3}$. Since $|M|$ has at
least two prime divisors, by Lemma 3.2, $\mathrm{o}(M)>\frac{13}{6}$, and so
$\mathrm{o}(G)>2.4$, which is a contradiction. Therefore, $n_{2}(G\setminus
M)>\frac{|G|}{3}$, and by Lemma 2.7, it follows that $M$ is nilpotent. Now
Lemma 2.5($i$) implies that $\mathrm{o}(M)>4$. Hence,
$\mathrm{o}(G)\geq\frac{\mathrm{o}(M)}{2}+1>3$, a contradiction.

Note that the previous lemma implies that $\mathrm{o}(G)<2$ if and only if $G$
is elementary abelian.

######  Remark 3.4.

Throughout this paper, for simplicity we say $G$ is a $\star$-group, if $G$ is
isomorphic to one of the following groups:

  1. it(i)

$D_{8}\times D_{8}$;

  2. it(ii)

$D(A)$, where $A$ is an abelian $2$-group;

  3. it(iii)

$H(r)$, for some integer $r$;

  4. it(iv)

$S(r)$, for some integer $r$.

######  Lemma 3.5.

Let $G$ be a $2$-group with $\mathrm{o}(G)<2.8$. Then
$n_{2}(G)>\frac{3}{5}|G|-2$, if $|G|\geq 16$.

###### Proof.

On the contrary assume that $n_{2}(G)\leq\frac{3}{5}|G|-2$. So,
$\mathrm{o}(G)\geq\frac{1+2n_{2}(G)+4(|G|-n_{2}(G)-1)}{|G|}\geq
2.8+1/|G|>2.8$, a contradiction.

######  Lemma 3.6.

Let $G$ be a $\star$-group, and $\mathrm{o}(G)<2.8$. Then $G$ is isomorphic to
one of the following groups:

  1. (i)

$H(2)$;

  2. (ii)

$S(2)$;

  3. (iii)

$D(C_{4}\times C_{4})$;

  4. (iv)

$D(C_{4}\times C_{2}^{k})$, for some integer $k\geq 0$.

###### Proof.

We consider each $\star$-group separately. Note that $\mathrm{o}(D_{8}\times
D_{8})=\frac{183}{64}>2.8$. If $G\cong D(A)$, where $A$ is an abelian
$2$-group, then $\mathrm{o}(G)=\frac{\mathrm{o}(A)}{2}+1$, and since
$\mathrm{o}(G)<2.8$, if follows that $\mathrm{o}(A)<3.6$. Since
$\mathrm{o}(C_{8})>\mathrm{o}(C_{4}^{3})>\mathrm{o}(C_{4}\times C_{4}\times
C_{2})>3.6$, we get that $A$ is isomorphic to $C_{4}\times C_{4}$ or
$C_{4}\times C_{2}^{k}$, for some integer $k\geq 0$. If $G\cong H(r)$, then
$|H(r)|=2^{2r+1}$, and $n_{2}(H(r))=2^{2r}+2^{r}-1$. Now by Lemma 3.5, $r\leq
2$. If $G\cong S(r)$, then $|S(r)|=2^{2r+1}$, and we can see that
$n_{2}(S(r))=2^{2r}+2^{r}-1$, which implies that $r\leq 2$, by Lemma 3.5. Note
that $S(1)\cong H(1)\cong D_{8}$.

Before we classify $2$-groups with $\mathrm{o}(G)<2.8$, we take a close look
at the bellow theorem, which is proved by Wall in [10, Pages 261-262]. It is
the key to classify such $2$-groups.

######  Theorem (C. T. C. Wall, [10]).

Let $G$ be a finite group. If $n_{2}(G)>\frac{1}{2}|G|-1$, then $G$ is
isomorphic to one of the following groups:

  1. it(a)

$D(A)$, where $A$ is an abelian group;

  2. it(b)

$D_{8}\times D_{8}\times C_{2}^{k}$, for some integer $k\geq 0$;

  3. it(c)

$H(r)\times C_{2}^{k}$, for some integers $r$ and $k\geq 0$;

  4. it(d)

$S(r)\times C_{2}^{k}$, for some integers $r$ and $k\geq 0$.

######  Theorem 3.7.

Let $G$ be a $2$-group with $\mathrm{o}(G)<2.8$. Then $G$ is isomorphic to one
of the following groups, where $k\geq 0$ is an integer:

  1. (i)

$C_{4}$;

  2. (ii)

$D_{8}\times C_{2}^{k}$;

  3. (iii)

$D(C_{4}\times C_{4})\times C_{2}^{k}$;

  4. (iv)

$(D_{8}*D_{8})\times C_{2}^{k}$;

  5. (v)

$S(2)\times C_{2}^{k}$;

  6. (vi)

an elementary abelian $2$-group.

###### Proof.

It is easy to check that the statement holds when $|G|<16$. Assume that
$|G|\geq 16$. If $G$ is abelian and $G$ has a factor isomorphic to
$C_{4}\times C_{2}$ or $C_{8}$, then $\mathrm{o}(G)>\mathrm{o}(C_{4}\times
C_{2})=2.875>2.8$, a contradiction. Obviously the result holds for elementary
abelian $2$-groups. Hence, assume that $G$ is non-abelian. By Lemma 3.5, we
get that $n_{2}(G)>\frac{3}{5}|G|-2$, and since $|G|\geq 16$,
$n_{2}(G)>\frac{1}{2}|G|-1$. Now by the above theorem, we see that $G$ is
isomorphic to one of the following $2$-groups:

  1. it(1)

$D(A)$, where $A$ is an abelian $2$-groups;

  2. it(2)

$D_{8}\times D_{8}\times C_{2}^{k}$, for some integer $k\geq 0$;

  3. it(3)

$H(r)\times C_{2}^{k}$, for some integers $r$ and $k\geq 0$;

  4. it(4)

$S(r)\times C_{2}^{k}$, for some integers $r$ and $k\geq 0$.

In case (1), the statement holds by Lemma 3.6. In other cases, let $E$ be the
direct factor isomorphic to $C_{2}^{k}$, and since
$\mathrm{o}(G/E)\leq\mathrm{o}(G)<2.8$, again by Lemma 3.6 and some easy
calculations we get the result.

Proof of the Main Theorem. Easily we can see that, if $G$ is isomorphic to one
of the groups listed in the Main Theorem, then $\mathrm{o}(G)<2.8$. Assume
that $G$ is a counterexample of minimal order. Remark that as
$\mathrm{o}(G)<\mathrm{o}(A_{5})$, by [3, Theorem C], $G$ is solvable. Let $N$
be a minimal normal $q$-subgroup of $G$, for some prime $q$. As
$\mathrm{o}(G/N)<\mathrm{o}(G)$ and there is no cyclic counterexample, $G/N$
satisfies the hypothesis of the theorem. So we consider each possibility for
$G/N$ separately:

($i$) Let $G/N\cong D_{12}$.

In this case, $G$ has a subgroup of index $2$, say $M$, such that $M$ has a
quotient isomorphic to $C_{6}$. So, $\mathrm{o}(M)>\mathrm{o}(C_{6})=3.5$. If
$n_{2}(G\setminus M)\leq|G|/3$, then
$\mathrm{o}(G)\geq\frac{\psi(M)+2|G|/3+4|G|/6}{|G|}>\frac{7}{4}+\frac{4}{3}>2.8$,
a contradiction. Hence, $n_{2}(G\setminus M)>|G|/3$, and by Lemma 2.7, $M$ is
nilpotent. So by Lemma 2.5, $\mathrm{o}(M)>4$, implying a contradiction by
Lemma 2.6.

($ii$) Let $G/N\cong S_{4}$.

In this case, by Lemma 2.1, $\mathrm{o}(G)\geq 2.75+\frac{\mathrm{o}(N)}{24}$.
Since $N$ is non-trivial, we get that $\mathrm{o}(N)\geq 1.5$, hence
$\mathrm{o}(G)>2.8$, a contradiction.

($iii$) Let $G/N\cong C_{3}^{a}$, where $a\in\\{1,2\\}$.

In this case, let $M$ be a normal subgroup of $G$ of index $3$. By Lemma 2.6,
$\mathrm{o}(M)<2.4$. Now by Lemma 3.3, $a=1$ and the only possibilities for
$M$ are $C_{3}$ and $C_{2}^{k}$, for some integer $k$. In the first case, as
$\mathrm{o}(C_{9})>6$, $G\cong C_{3}^{2}$, we get a contradiction. In the
second case, $G$ is a group of order $3\cdot 2^{k}$ and Fitting lemma implies
that $N=C_{N}(P)\times[P,N]$, where $P\in\mathrm{Syl}_{3}(G)$. By the fact
that $N$ is a minimal normal subgroup of $G$ and $C_{N}(P)\triangleleft G$, we
get that either $G$ is a Frobenius group described in Case ($iv$), which is
impossible as $G$ is a counterexample, or $G=P\times N$, a contradiction by
Lemma 2.5($i$).

($iv$) Let $G/N\cong C_{2}^{2k}\rtimes C_{3}$, be a Frobenius group, for some
integer $k$.

If $N$ is a $2$-group, then $N\leq{\bf Z}(P)$, where $P$ is the Sylow
$2$-subgroup of $G$. Note that by Lemma 2.6, $\mathrm{o}(P)<2.4$, and by Lemma
3.3, we get that $P\cong D_{8}$ or $P\cong C_{2}^{a}$, for some integer $a$ .
Note that as ${\mathrm{Aut}}(D_{8})\cong D_{8}$, the first case implies the
nilpotency of $G$, a contradiction by Lemma 2.5. Therefore, $P\cong
C_{2}^{a}$. First, assume that $N\leq{\bf Z}(G)$, then $|N|=2$. Thus,
$n_{6}(G\setminus P)\geq n_{3}(G\setminus P)$, and

$\mathrm{o}(G)\geq\frac{\psi(P)+3(|G|-|P|)/2+6(|G|-|P|)/2}{3|P|}=\frac{\mathrm{o}(P)}{3}+3>3,$

a contradiction. So, $N\cap{\bf Z}(G)=1$ and we get that the Sylow
$3$-subgroup of $G$ acts on $N$ and $G/N$, Frobeniusly. Therefore, $G$ is a
Frobenius group with an elementary abelian $2$-group as its kernel, which is
the group described in Case ($iv$), a contradiction. Now, we assume that $|N|$
is odd. In this case, using Lemma 2.6, $\mathrm{o}(M)<2.4$, where $M$ is an
index $3$ normal subgroup of $G$. So, by Lemma 3.3, $M\cong D(C_{3}\times
C_{3})$, which implies $2k=1$, a contradiction.

($v$) Let $G/N$ be a $2$-group.

Note that by Theorem 3.7, $G$ is not a $2$-group. Now by Lemma 3.1, $|G/N|\geq
8$. We claim that $G/G^{\prime}N$ is an elementary abelian $2$-group.
Otherwise, there exists a normal subgroup $H$ of $G$ such that $G/H\cong
C_{4}$,
$\mathrm{o}(G)\geq\frac{\psi(H)+10|H|}{4|H|}=\frac{\mathrm{o}(H)}{4}+2.5$,
which implies that $\mathrm{o}(H)<1.2$, a contradiction. So, $G/G^{\prime}N$
is an elementary abelian $2$-group. First, let $G^{\prime}N=N$. Then every
subgroup of order $2|N|$ is a normal subgroup of $G$. Let $K$ be such a
subgroup. Then the Sylow $q$-subgroup of ${\bf Z}(K)$ is normal in $G$ and by
the fact that $N$ is a minimal normal subgroup, either $K\cong N\times M$,
where $M$ is a group of order $2$, or $K$ is a Frobeniuos group. If the first
case occurs, by the minimality of $|G|$, $G/M$ is isomorphic to $D_{12}$. Note
that since $G/N$ is completely reducible, $G$ splits on $M$. Therefore,
$G\cong C_{2}\times D_{12}$, which is a contradiction, since
$\mathrm{o}(C_{2}\times D_{12})=\frac{73}{24}>3$. So, $K$ is a Frobenius
group. Since this holds for all subgroups of order $2|N|$, we get that $G$ is
a Frobenius group and $G/N\cong C_{2}$, a contradiction. Therefore,
$N<G^{\prime}N$. Let $S$ be a normal subgroup of $G$ containing $N$, such that
$G^{\prime}N/S$ is a chief factor of $G$, isomorphic to $C_{2}$. Therefore,
$G/S$ is a generalized extraspecial group (see [8]), and by Theorem 3.7, we
conclude that $G/S\cong D_{8}\times C_{2}^{k}$ or $G/S\cong D_{8}*D_{8}\times
C_{2}^{k}$, for some $k\geq 0$. In the first case, $G$ has a factor $G/L$
isomorphic to $D_{8}$. Let $T/L$ be a subgroup of $G/L$ such that $T/L\cong
C_{4}$. Note that $N\leq L$, hence, $\mathrm{o}(L)>2$. Therefore,
$\mathrm{o}(T)\geq\frac{\psi(L)+10|L|}{4|L|}=2.5+\frac{\mathrm{o}(L)}{4}>3$.
If $n_{2}(G\setminus T)\leq|G|/3$,
$\mathrm{o}(G)\geq\frac{\psi(T)+2|G|/3+4|G|/6}{|G|}=\frac{\mathrm{o}(T)}{2}+\frac{4}{3}>2.8$,
a contradiction. So, $n_{2}(G\setminus T)>|G|/3$, by Lemma 2.7, $T$ is
nilpotent. Then, by Lemma 2.5($i$), $\mathrm{o}(T)>4$, which implies a
contradiction by Lemma 2.6. In the second case, there exists a subgroup $W\leq
G$ of index $2$, where $W$ has a quotient, say $W/H$, isomorphic to
$D_{8}*C_{4}$. Note that $\mathrm{o}(D_{8}*C_{4})=\frac{47}{16}=2.9375$, and
since $N\leq H$, $\mathrm{o}(H)>2$. By Lemma 2.1, we get that
$\mathrm{o}(W)>3$. Now similar to the previous case, we get a contradiction.

($vi$) Let $G/N\cong C_{3}^{k}\rtimes C_{2}$ be a Frobenius group, for some
integer $k$.

By assumption, $|N|=q^{a}$, for some integer $a$. If $q$ is odd, then by Lemma
3.1($i$), $q=3$ and $G\cong C_{3}^{a+k}\rtimes C_{2}$, which is a Frobenius
group, a contradiction. So, $q=2$. Note that every subgroup $M$ containing $N$
of order $3|N|$ is a normal subgroup of $G$. Hence, the Sylow $2$-subgroup of
${\bf Z}(M)$ is a normal subgroup of $G$, and as $N$ is minimal normal in $G$,
either ${\bf Z}(M)=M$ or ${\bf Z}(M)=1$. In the first case, $M=N\times Q$,
where $Q$ is the Sylow $3$-subgroup of $M$. By the above discussion, $G/Q$ is
not isomorphic to the groups stated in Cases ($i$)-($v$), and since
$4\mid|G|$, we get that $G/Q$ is not isomorphic to the group mentioned in Case
($vi$), a contradiction to the minimality of $|G|$. Whence, ${\bf Z}(M)=1$,
which yields that $M$ is a Frobenius group. Thus, there is no element of order
$6$ in $G$, implying that there is a normal subgroup of $G$ of index $2$, say
$T$, which is a Frobenius group and by the structure of the Frobenius groups,
we get that $k=1$. So, Syl${}_{2}(G)=\left\\{P_{1},P_{2},P_{3}\right\\}$, and
$N=P_{i}\cap P_{j}$, for $1\leq i<j\leq 3$. If the Sylow 2-subgroups of $G$
are abelian, then $\bigcup\limits_{i=1}^{3}{P_{i}}\subset C_{G}(N)$, hence,
$C_{G}(N)=G$, a contradiction. Whence, $|G|\geq 24$. Sylow $2$-subgroups of
$G$ are not abelian, so by [10, Corollary], every Sylow $2$-subgroup of $G$
has at least $|G|/12$ elements of order $4$. On the other hand, we know that
$n_{3}(G)=|G|-|P_{1}\cup P_{2}\cup P_{3}|=|G|/3$. Hence,
$\psi(G)=1+3n_{3}(G)+4n_{4}(G)+2(|G|-n_{3}(G)-n_{4}(G)-1)\geq\frac{17}{6}|G|-1$.
Hence, $\mathrm{o}(G)\geq\frac{17}{6}-\frac{1}{|G|}$, which implies that
$|G|=24$, and so $G\cong S_{4}$, a contradiction.

######  Remark 3.8.

All the groups in the Main Theorem and their average orders are listed in the
following tables:

Groups |  $\psi(G)$ |  $\mathrm{o}(G)$
---|---|---
$C_{3}^{k}\rtimes C_{2}$ |  $5\cdot 3^{k}-2$ |  $2.5-1/3^{k}$
$C_{3}$ |  $7$ |  $7/3$
$C_{2}^{2k}\rtimes C_{3}$ |  $2^{2k+3}-1$ |  $8/3-1/(3\cdot 2^{2k})$
$D_{12}$ |  $33$ |  $2.75$
$C_{3}\times C_{3}$ |  $25$ |  $25/9$
$S_{4}$ |  $67$ |  $67/24$
(a) Non 2-groups with $\mathrm{o}(G)<2.8$

Groups |  $\psi(G)$ |  $\mathrm{o}(G)$
---|---|---
$C_{2}^{k}$ |  $2^{k+1}-1$ |  $2-1/2^{k}$
$D_{8}\times C_{2}^{k}$ |  $5\cdot 2^{k+2}-1$ |  $2.5-1/2^{k+3}$
$D(C_{4}\times C_{4})\times C_{2}^{k}$
$(D_{8}*D_{8})\times C_{2}^{k}$
$S(2)\times C_{2}^{k}$ |
$11\cdot 2^{k+4}-1$ |
$11/4-1/2^{k+5}$
$C_{4}$ |  $11$ |  $2.75$
(b) $2$-Groups with o($G$)$<2.8$

Proof of the Corollary

($a$) Let $\mathrm{o}(H)=\mathrm{o}(S_{4})=\frac{67}{24}$. Then $|H|=24m$, for
some odd integer $m$. Therefore, $H$ is not a $2$-group. Now using the Main
Theorem, since $24\mid|G|$ and $\mathrm{o}(H)<2.8$, we get that $G\cong
S_{4}$. If $\mathrm{o}(H)=\mathrm{o}(G)$, where $G$ is one of the non
$2$-groups mentioned in ($a$), similarly we get the result. If
$\mathrm{o}(H)=\mathrm{o}(D_{8}\times C_{2}^{k})=(5\cdot 2^{k+2}-1)/2^{k+3}$,
for some integer $k$, then $|G|=2^{k+3}m$, for some odd integer $m$. If $m\geq
3$, then $H$ is isomorphic to one of the groups in Table A. However, $(5\cdot
2^{k+2}-1)/2^{k+3}$ is not equal to the average orders in Table A, a
contradiction. So, $m=1$, and $H$ is a $2$-group of order $2^{k+3}$ with
$\mathrm{o}(H)<2.8$. In Table B, we see that $H\cong D_{8}\times C_{2}^{k}$.
Hence, $D_{8}\times C_{2}^{k}$ is characterizable by average order. The same
discussion shows the charactrizablity of $C_{2}^{k}$ by average order.

($b$) Note that $\mathrm{o}(D_{12})=\mathrm{o}(C_{4})=2.75$. Now assume that
$\mathrm{o}(H)=\frac{11}{4}=2.75$. We get that $|G|=4m$, for some odd integer
$m$. If $m=1$, as $\mathrm{o}(C_{2}\times C_{2})=1.75$, $H\cong C_{4}$.
Otherwise, since $\mathrm{o}(H)<2.8$, by comparing the average orders of the
groups in Table A, we see that $H\cong D_{12}$.

($c$) Remark that $\mathrm{o}(D(C_{4}\times C_{4})\times
C_{2}^{k})=\mathrm{o}((D_{8}*D_{8})\times C_{2}^{k})=\mathrm{o}(S(2)\times
C_{2}^{k})$, for any integer $k$. Assume that $\mathrm{o}(H)=(11\cdot
2^{k+4}-1)/2^{k+5}$, for some integer $k$. We see that $|G|=2^{k+5}m$, for
some odd integer $m$. If $m\geq 3$, then as there is no group with such
average order in Table A, we get a contradiction. Therefore, $m=1$, and $H$ is
a $2$-group of order $2^{k+5}$ with $\mathrm{o}(H)<2.8$. Now Table B shows
that $H$ is isomorphic to one of three mentioned groups.

According to the above corollary we pose the following two questions:

######  Question 3.9.

What are the values of $n$ for which there exists a $n$-recognizable group by
average order?

######  Question 3.10.

Is there any finite group $G$ such that there exists infinitely many non-
isomorphic groups with average order $\mathrm{o}(G)$?

As an application, by calculating the average orders in the Main Theorem, we
get that other than $S_{4}$, there exists no group $G$, such that
$\mathrm{o}(G)$ lies in the interval $[\frac{67}{24},2.8]$. This is a partial
answer to [6, Conjecture 2.11], about the density of
$Im(\mathrm{o})=\left\\{\mathrm{o}(G)|G\text{ is a finite group}\right\\}$.
Now we see that $Im(\mathrm{o})$ is not dense in $[a,\infty)$, for any
$a\leq\frac{67}{24}$.

Conflict of Interest he authors have no conflicts of interest to declare. All
co-authors have seen and agree with the contents of the manuscript and there
is no financial interest to report.

## References

  * [1] H. Amiri, S.M. Jafarian Amiri, I.M. Isaacs, Sums of element orders in finite groups, Comm. Algebra 37 (2009), 2978-2980
  * [2] D. Gorenstein, Finite groups, Chelsea Pub Co, New York (1980).
  * [3] M. Herzog, P. Longobardi and M. Maj, Another criterion for solvability of finite groups, J. Algebra, 597 (2022), 1-23.
  * [4] M. Herzog, P. Longobardi and M. Maj, On groups with average element orders equal to the average order of alternating group of degree 5, Glasnik Matematicki, (2023)
  * [5] E. I. Khukhro, A. Moreto and M. Zarrin, The average element order and the number of conjugacy classes of finite groups, J. Algebra 569 (2021), 1-11.
  * [6] M. S. Lazorec and M. Tarnauceanu, On the average order of a finite group, Journal of Pure and Applied Algebra, 227 (2023), article ID 107276.
  * [7] W. M. Potter, Nonsolvable groups with an automorphism inverting many elements, Arch. Math., 50 (1988), 292-299
  * [8] R. Stancu, Almost all generalized extraspecial $p$-groups are resistant, J. Algebra, 249 (2002), 120-126
  * [9] M. Tarnauceanu, Another criterion for supersolvability of finite groups, J. Algebra, 604 (2022), 683-692.
  * [10] C. T. C. Wall, On groups consisting mostly of involutions, Math. Proc. Camb. Philos. Soc. 67 (1970), 251-262
  * [11] The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.12.2, 2022, https://www.gap-system.org/Releases/4.12.2.html


✓✓ ✗✗

# CattleEyeView: A Multi-task Top-down View Cattle Dataset for Smarter
Precision Livestock Farming

Kian Eng Ong1‡ Sivaji Retta2‡ Ramarajulu Srinivasan2‡@ Shawn Tan2 Jun Liu1 1
Information Systems Technology and Design, Singapore University of Technology
and Design, Singapore
2 AnimalEYEQ Private Limited, Singapore

###### Abstract

Cattle farming is one of the important and profitable agricultural industries.
Employing intelligent automated precision livestock farming systems that can
count animals, track the animals and their poses will raise productivity and
significantly reduce the heavy burden on its already limited labor pool. To
achieve such intelligent systems, a large cattle video dataset is essential in
developing and training such models. However, many current animal datasets are
tailored to few tasks or other types of animals, which result in poorer model
performance when applied to cattle. Moreover, they do not provide top-down
views of cattle. To address such limitations, we introduce CattleEyeView
dataset, the first top-down view multi-task cattle video dataset for a variety
of inter-related tasks (i.e., counting, detection, pose estimation, tracking,
instance segmentation) that are useful to count the number of cows and assess
their growth and well-being. The dataset contains 753 distinct top-down cow
instances in 30,703 frames (14 video sequences). We perform benchmark
experiments to evaluate the model’s performance for each task. The dataset and
codes can be found at https://github.com/AnimalEyeQ/CattleEyeView

###### Index Terms:

dataset, cow, detection, tracking, counting, pose estimation, instance
segmentation

\textsuperscript{\textdaggerdbl}\textsuperscript{\textdaggerdbl}footnotetext:
Equal contributions. @Corresponding author<EMAIL_ADDRESS>
##  I Introduction

Monitoring and assessing the health, well-being, growth, and behaviors of
animals are especially important in the livestock farming industry, as all of
these ultimately impact farm production and profits [28]. Cattle farming is
one of the important and profitable agricultural industries, especially in the
United States [10]. Cows are raised for their meat as well as milk which is
then further processed to dairy products (e.g., cheese, butter, yogurt).
Globally, the demand for cattle is expected to increase by 5.8% in the next 10
years [2]. Given the strain on the farming industry with an increasing number
of animals but decreasing number of farmers [5], employing automated
Artificial Intelligence (AI) systems in the livestock farming industry will
raise productivity and expedite the industry’s effort in digital
transformation towards smart precision livestock farming while significantly
reducing the heavy burden on its already limited labor pool [22, 12].
Precision livestock farming includes the use of intelligent systems to count
animals, track and analyze poses and movements of animals and their heads,
bodies, and limbs, as well as their activities and behaviors in the farm [21,
5].

Accurate counting of livestock is particularly important in the livestock
industry as each farm on average sells up to 5000 cows each day [8]. Counting
errors are common when manually counted by humans and are substantially
costly, given that each cow is priced at around USD$1000 [8].

Besides counting, tracking each animal in the farm is precursor to evaluating
its state of health and analyzing its behaviors and interactions with others
[27, 30]. Additionally, detecting its keypoints is important in many practical
application scenarios such as animal identification, tracking, behavioral
analysis (including lameness detection), part segmentation, and even gait
analysis [4, 14]. On top of keypoints identification, segmentation can help to
chart and assess the animal’s well-being (e.g., cleanliness and weight) [13].
All of these interrelated tasks are crucial to achieve a comprehensive
understanding and analysis of the animal’s health and well-being. To develop
intelligent systems for smart precision livestock farming, a large video
dataset is essential in developing and training such models. However, there is
a lack of sufficient detailed annotations of animal videos that can be used
for training [14], which currently impedes research in this field.

Although there are existing animal datasets, some contain a diversity of
animals (e.g., Animal Pose [7], AP10K [29], Animal Kingdom [19]) while others
are for specific types of animals (e.g., sheep [20], and pigs [22]). Models
that are developed on such datasets are unable to achieve good performance
when applied to cattle. Moreover, most existing animal datasets provide
frontal and side views of the animals, but not the top-down views of animals.
The advantages of using the top-down view, instead of frontal or side views,
of the animal are (1) the animals are usually not severely occluded by other
farm objects or animals, (2) the appearances of animals are more consistent
(similar size and shape) as compared to isometric angled view that is commonly
seen in CCTV footages [22], (3) the top-down view focuses on a particular
animal, unlike isometric view which may contain differently sized animals in
the background that will distract the model. This is important to achieve
accurate counting of animals as well as taking into account the direction of
movement of the animal for accurate counting. All of these advantages are
important to accurately assess the growth and well-being of the animal.
However, models that are developed on existing datasets based on frontal, side
or isometric views of animals are unable to achieve good performance when
applied to situations like ours that require the top-down views of cattle.
There is only one cattle dataset [11] with a top-down view that only localizes
and re-identifies 186 cattle based on their black-and-white coat patterns
viewed from top-down.

Hence, to address this limitation, we construct CattleEyeView, the first
multi-task video cattle dataset using the bird’s-eye views (i.e., top-down
views) of 6 different cattle breeds with different coat colors. Our
CattleEyeView dataset not only provides carefully labeled annotations of (1)
bounding boxes of the cow’s body and head for cattle counting and tracking, we
also provide detailed annotations for (2) tracking ID for tracking, (3) cattle
count (i.e., number of cattle that have passed through the ramp) in the video
sequence for cattle counting, as well as (4) 24 keypoints for pose estimation,
and (5) segmentation masks for instance segmentation. Our CattleEyeView
dataset with various types of detailed annotations shall facilitate the
community and industry to develop, adapt, and evaluate various types of
advanced models, thereby promoting the development and deployment of smarter
precision livestock farming AI systems to fulfill diverse industry needs and
application scenarios.

##  II Related Works

Although there are existing animal datasets, some contain a diversity of
animals (e.g., Animal Pose [7], AP10K [29], Animal Kingdom [19]) from which
the models are developed on but often do not achieve good performance when
applied to specific types of farm animals. On the other hand, most animal
datasets are catered for specific task (e.g., pose estimation only [29]),
specific types of animals (e.g., sheep [20], and pigs [22]), or specific
environments (e.g., mice in laboratories [25]). Unlike controlled environments
in a laboratory, farms often have non-uniform background (e.g., dirt ground)
and sometimes with varying lighting conditions (especially outdoors).
Furthermore, the low contrast between the animal (e.g., cow) and the dirt
ground, as well as occlusion of the animals by other animals or even equipment
used in the farm, also pose practical challenges.

Moreover, most existing animal datasets provide frontal and side views of the
animals, but not the top-down views of animals. More crucially, models that
are trained on existing animal datasets with frontal, side or isometric views
of animals do not generalize well on top-down view of animals, because of the
difference in viewpoint. Therefore, there is a need for such top-down view
datasets in order to develop advanced intelligent automated system to cater
for different needs in precision livestock farming. However, there is only one
cattle dataset with a top-down view. Cows2021 [11] only localizes and re-
identifies 186 cattle based on their black-and-white coat patterns viewed from
top-down. As it primarily localizes and re-identifies the cattle, it does not
contain annotations for head bounding box, pose, and counting.

Different from existing animal datasets as shown in Table I, our CattleEyeView
video dataset not only provides carefully labeled annotations of bounding
boxes for cattle tracking and counting, our annotations also include keypoints
for pose estimation, and segmentation masks for instance segmentation.
Therefore, our dataset containing six different cow breeds with different coat
colors in different illumination conditions provides a challenging benchmark
for the research community and industry to develop, adapt, and evaluate
various types of advanced methods for smarter precision livestock farming.

TABLE I: Comparison with other cattle datasets

Dataset |  Publicly available? | Task | # frames | # keypoints | Type of view
---|---|---|---|---|---
Detection |  Counting |  Tracking |  Pose |  Segmentation
Cows2021 [11] | ✓ | ✓ | ✗ | ✗ | ✗ | ✗ | 10,402 | N.A. | Top
Deep cascaded convolutional models for cattle pose estimation [17] | ✗ | ✗ | ✗ | ✗ | ✓ | ✗ | 2134 | 16 | Side
Video-based multi-scene cow pose estimation [28] | ✗ | ✗ | ✗ | ✓ | ✓ | ✗ | 3000 | 16 | Side
Multicow pose estimation based on keypoint extraction [12] | ✗ | ✓ | ✗ | ✗ | ✓ | ✗ | 1800 | 16 | Side
CoWalk-30 [23] | ✗ | ✗ | ✗ | ✗ | ✓ | ✗ | 4236 | 17 | Side
CattleEyeView (Ours) | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | 30,703 | 24 | Top

##  III CattleEyeView Dataset

###  III-A Video Data Collection and Annotation

The 14 video sequences (containing 30,703 frames) containing top-down views of
cows are collected in the morning and evening over 4 months (2021-11-09 to
2022-03-04) from overhead RGB CCTV cameras fixed at a height of 3.7m with cows
walking through the outdoor loading ramp of a farm (Fig. 1). There cows have
different coat colors and patterns, and are seen in different illumination
conditions (Fig. 2). In total, there are 753 distinct cow instances which are
manually annotated by experienced human annotators and subsequently verified
by more senior annotators.

Figure 1: The camera is mounted at a height of 3.7m to capture the top-down
views of cattle. Figure 2: The video sequences contain different colored cows
found in different lighting conditions (different times and different days).

An example of the annotation can be seen in Fig. 3. We annotate the body and
head bounding boxes, as well as the segmentation mask of the cows. We also
track the cows by assigning a unique tracking ID and count the number of cows
as they walk through the loading ramp of a farm.

Additionally, we annotate 24 keypoints, together with the respective
visibility tag (i.e., visible and occluded), for the (1) head, (2) nose, (3)
left eye, (4) right eye, (5) base of left ear, (6) tip of left ear, (7) base
of right ear, (8) tip of right ear, (9) neck, (10) withers, (11) left of front
elbow, (12) left of front knee, (13) left of front paw, (14) right of front
elbow, (15) right of front knee, (16) right of front paw, (17) left of back
elbow, (18) left of back knee, (19) left of back paw, (20) right of back
elbow, (21) right of back knee, (22) right of back paw, (23) base of tail, and
(24) end of tail. The head keypoints (1 to 10) and rear keypoints (23 and 24)
are chosen to reflect the where the cow is facing, and also take into account
when some keypoints may be occluded. The limb keypoints (11 to 22) are
landmarks to indicate how the cow moves, as well as facilitate the assessment
of its growth (including weight and gait).

Figure 3: Example of annotations in a frame of the video sequence: Bounding
boxes of the head and body of the cattle, as well as keypoints and
segmentation mask of the cattle. Other annotations not shown in the image
above include tracking ID and count number of cattle that have passed through
the ramp.

###  III-B Dataset Description

The CattleEyeView dataset contains 753 distinct top-down cow instances in
30,703 frames (14 video sequences) with (1) bounding boxes of the cow’s body
and head for cattle counting and tracking, we also provide detailed
annotations for (2) tracking ID for tracking, (3) cattle count (i.e., number
of cattle that have passed through the ramp) in the video sequence for cattle
counting, as well as (4) 24 keypoints for pose estimation, and (5)
segmentation masks for instance segmentation

Top-down cattle view. The top-down views of animals are usually not severely
occluded by farm objects or other animals. Additionally, the appearances of
animals in top-down view are more consistent (similar size and shape) than
isometric view that is commonly seen in CCTV footages. Moreover, the top-down
view focuses on a particular animal, unlike isometric view which may contain
differently sized animals in the background that distracts the model. Hence,
these advantages are beneficial in accurately counting the animals and
analyzing their growth and well-being.

Different cow breeds and different coat colors. Our dataset contains images of
six different cow breeds (Angus, Brahman, Charolais, Droughtmaster, Hereford,
and Santa Gertrudis). These cows have different coat colors. Having a
diversity of cow breeds and coat colors improves the generalization ability of
the models to detect the cows and perform various downstream tasks.

Various illumination conditions. The footages are obtained at different times
of days, with different illumination conditions. The shadow cast on the cows
sometimes results in low foreground-background contrast, thereby introducing
significant and practical challenges such as identifying and delineating the
cows and the dirt ground.

###  III-C Dataset Tasks and Annotations

Object detection. In detection, the model takes in the image of the animal and
outputs the spatial location (bounding box) of the animal in the frame (Fig.
4).

Figure 4: In object detection (top-right image), the model identifies and
outputs the spatial location of each animal. In pose estimation (bottom-left
image), the model predicts all the keypoints of each animal. In instance
segmentation (bottom-right image), the model outputs the binary segmentation
mask of each animal.

Tracking and counting. In tracking, the model takes in the video of the animal
and predicts the trajectory of the animal (i.e., bounding box across frames)
(Fig. 5). For counting, the model takes in the video of the animal and outputs
the number of animals that have crossed a pre-defined region in the frame of
the video (Fig. 5).

Figure 5: In tracking, the model identifies and tracks the movement of each
animal.

Pose estimation. In pose estimation, the model takes in the image of the
animal and predicts the different keypoints (e.g., joints) on the animal (Fig.
4).

Instance segmentation. In instance segmentation, the model takes in the image
of the animal and outputs the binary segmentation mask of each animal, which
outlines each instance of the animal (Fig. 4).

In a nutshell, our CattleEyeView dataset contains various types of annotations
that will be useful to train various types of models for smarter precision
livestock farming. Our dataset provides a challenging benchmark for the
research community and industry to develop, adapt, and evaluate various types
of advanced methods for smarter precision livestock farming.

##  IV Experiments

To facilitate training for various tasks, each video is either assigned to the
train set (75%) or test set (25%). We use YOLOv5 [15], YOLOv8 [16], OpenMMLab
[9], and BoxMOT [6] code libraries to evaluate state-of-the-art methods based
on respective existing established evaluation metrics on our dataset (YOLOv8
[16] for cattle detection, BoxMOT using YOLOv8 with ByteTrack [31] and YOLOv8
with BoT-SORT [3] for tracking and counting, HRNet [24] and YOLOv8-Pose [16]
for pose estimation, YOLOv5-Segment [15] and YOLOv8-Segment [16] for instance
segmentation).

Table II shows the results of the body and head detection. The models achieve
high mAP0.5 scores for both body and head, despite color variations of the
cows and different lighting conditions. The head detection works better than
body detection, thus suggesting that further works are needed to improve the
body detection. In comparison, the models achieve better performance for
Cows2021 dataset, hence suggesting that advanced methods are needed to improve
performance on our dataset that is more challenging.

TABLE II: Results of cattle detection

Dataset | Model | Body | Head
---|---|---|---
Precision↑ |  Recall↑ |  mAP0.5↑ |  mAP0.5:0.95↑ |  Precision↑ |  Recall↑ |  mAP0.5↑ |  mAP0.5:0.95↑
Cows2021 | YOLOv8 [16] | 0.960 | 0.962 | 0.987 | 0.904 | N.A. | N.A. | N.A. | N.A.
Cows2021 | RT-DETR[18] | 0.981 | 0.950 | 0.989 | 0.845 | N.A. | N.A. | N.A. | N.A.
Ours | YOLOv8 [16] | 0.924 | 0.888 | 0.956 | 0.858 | 0.816 | 0.712 | 0.809 | 0.570
Ours | RT-DETR[18] | 0.917 | 0.809 | 0.877 | 0.678 | 0.876 | 0.662 | 0.720 | 0.465

Table III shows the results of tracking evaluated on a series of metrics, such
as Precision, Recall, Multiple Object Tracking Accuracy (MOTA) and Precision
(MOTP) that are used in MOT-Challenge [26] evaluation metrics, as well as
Identification Precision (IDP), Recall (IDR), and F1 score (IDF1). The
tracking models use the state-of-the-art YOLOv8 [16] as the detection model
and are evaluated using py-motmetrics [1]. The models achieve good MOTA and
MOTP tracking scores, since the body detection models have already achieved
good performance as seen from the high IDP, IDR, and IDF1 scores. The table
also shows the accuracy of cow counts that is evaluated using Mean Absolute
Error (MAE).

TABLE III: Results of cattle tracking and counting

Model |  Precision↑ |  Recall↑ |  MOTA↑ |  MOTP↑ |  IDF1↑ |  IDP↑ |  IDR↑ |  MAE↓
---|---|---|---|---|---|---|---|---
BoTSORT [3] | 0.211 | 0.036 | 0.167 | 0.312 | 0.025 | 0.087 | 0.015 | 20.5
ByteTrack [31] | 0.201 | 0.034 | 0.170 | 0.336 | 0.025 | 0.091 | 0.015 | 24.0

While the cattle detection obtains promising results, it is still challenging
to track the cow and obtain accurate count of the cows as seen in the low
metric scores in Table III and high MAE w.r.t. average of 58 cows in the
videos (i.e., 41% error) respectively. Tracking the cows is challenging as
they may move back-and-forth across the tracking line, leading to multiple
counts of the same cow. Also, the low scores for pose estimation in Table IV
is likely the result of the challenging top-down views. Hence, more advanced
algorithms need to be developed to address these challenges.

TABLE IV: Results of cattle pose estimation.

Model | OKS=0.5
---|---
AP↑ | AR↑ | mAP↑
HRNet [24] | 0.459 | 0.615 | 0.122
YOLOv8 Pose [16] | 0.638 | 0.348 | 0.338

OKS, AP, AR, mAP refer to Object Keypoint Similarity, Average Precision,
Average Recall, mean Average Precision respectively.

The results in Table V shows that the models obtain high mAP0.5 scores and are
able to accurately segment the cows from the dirt background.

TABLE V: Results of instance segmentation Model | Precision↑ | Recall↑ | mAP0.5↑ | mAP0.5:0.95↑
---|---|---|---|---
YOLOv5 [15] | 0.877 | 0.774 | 0.875 | 0.661
YOLOv8 [16] | 0.853 | 0.773 | 0.849 | 0.631

In summary, it can be observed that while good performance can be achieved for
detection and segmentation, there remains a gap to achieve better performance.
Moreover, it is still challenging to track the cow, estimate its keypoints,
and obtain an accurate count. Thus, this shall motivate the community to
develop, adapt, and evaluate various types of more advanced methods to tackle
these challenges.

##  V Conclusion

We construct CattleEyeView, the first top-down view and multi-task cattle
video dataset for a diversity of inter-related tasks (i.e., counting,
detection, pose estimation, tracking, instance segmentation) that are useful
to count the number of cows and assess the cow’s growth and well-being. We
perform benchmark experiments to evaluate the performance of models for each
task. We hope that this dataset will facilitate the community and industry to
develop, adapt, and evaluate various types of advanced methods to fulfill
diverse needs and application scenarios for smarter precision livestock
farming.

##  VI Acknowledgments

This work was supported by the National Research Foundation Singapore under
its AI Singapore Programme (Award number: Grant AISG-100E-2020-065). We would
like to thank Coggan Farms for providing the video footages and making these
footages available for research purposes, as well as our annotators at
Keylabs.ai for annotating the videos and checking the quality of the
annotations.

## References

  * [1] Benchmark multiple object trackers (MOT) in Python.  https://github.com/cheind/py-motmetrics.
  * [2] OECD-FAO Agricultural Outlook 2021-2030, 2021.
  * [3] Nir Aharon, Roy Orfaig, and Ben-Zion Bobrovsky.  BoT-SORT: Robust Associations Multi-Pedestrian Tracking.  arXiv preprint arXiv:2206.14651, 2022.
  * [4] Prianka Banik, Lin Li, and Xishuang Dong.  A Novel Dataset for Keypoint Detection of quadruped Animals from Images.  ArXiv, abs/2108.13958, 2021.
  * [5] Daniel Berckmans.  General introduction to precision livestock farming.  Animal Frontiers, 7(1):6–11, 2017.
  * [6] Mikel Broström.  BoxMOT: A collection of SOTA real-time, multi-object trackers for object detectors .  https://github.com/mikel-brostrom/yolo_tracking.
  * [7] Jinkun Cao, Hongyang Tang, Hao-Shu Fang, Xiaoyong Shen, Cewu Lu, and Yu-Wing Tai.  Cross-domain adaptation for animal pose estimation.  In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 9498–9507, 2019.
  * [8] Garze Clarissa.  Cattle-ist for the Future: Plainsight Revolutionizes Livestock Management with AI, 2021.
  * [9] OpenMMLab Contributors.  OpenMMLab.  https://github.com/open-mmlab, 2020.
  * [10] United States Department of Agriculture Economic Research Service.  Sector at a Glance: Beef and Cattle, 2022.
  * [11] Jing Gao, Tilo Burghardt, William Andrew, Andrew W Dowsey, and Neill W Campbell.  Towards Self-Supervision for Video Identification of Individual Holstein-Friesian Cattle: The Cows2021 Dataset.  arXiv preprint arXiv:2105.01938, 2021.
  * [12] Caili Gong, Yong Zhang, Yongfeng Wei, Xinyu Du, Lide Su, and Zhi Weng.  Multicow pose estimation based on keypoint extraction.  PloS one, 17(6):e0269259, 2022.
  * [13] Nan Jia, Gert Kootstra, Peter Groot Koerkamp, Zhengxiang Shi, and Songhuai Du.  Segmentation of body parts of cows in RGB-depth images based on template matching.  Computers and Electronics in Agriculture, 180:105897, 2021.
  * [14] Le Jiang, Caleb Lee, Divyang Teotia, and Sarah Ostadabbas.  Animal pose estimation: A closer look at the state-of-the-art, existing gaps and opportunities.  Computer Vision and Image Understanding, page 103483, 2022.
  * [15] Glenn Jocher.  YOLOv5 by Ultralytics.  https://github.com/ultralytics/yolov5, May 2020.
  * [16] Glenn Jocher, Ayush Chaurasia, and Jing Qiu.  YOLO by Ultralytics.  https://github.com/ultralytics/ultralytics, Jan. 2023.
  * [17] Xiangyuan Li, Cheng Cai, Ruifei Zhang, Lie Ju, and Jinrong He.  Deep cascaded convolutional models for cattle pose estimation.  Computers and Electronics in Agriculture, 164:104885, 2019.
  * [18] Wenyu Lv, Shangliang Xu, Yian Zhao, Guanzhong Wang, Jinman Wei, Cheng Cui, Yuning Du, Qingqing Dang, and Yi Liu.  Detrs beat yolos on real-time object detection, 2023.
  * [19] Xun Long Ng, Kian Eng Ong, Qichen Zheng, Yun Ni, Si Yong Yeo, and Jun Liu.  Animal Kingdom: A Large and Diverse Dataset for Animal Behavior Understanding.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 19023–19034, June 2022.
  * [20] Kehinde Owoeye and Stephen Hailes.  Online collective animal movement activity recognition.  arXiv preprint arXiv:1811.09067, 2018.
  * [21] Mahir Patel, Yiwen Gu, Lucas C Carstensen, Michael E Hasselmo, and Margrit Betke.  Animal Pose Tracking: 3D Multimodal Dataset and Token-based Pose Optimization.  International Journal of Computer Vision, 131(2):514–530, 2023.
  * [22] Eric T Psota, Mateusz Mittek, Lance C Pérez, Ty Schmidt, and Benny Mote.  Multi-pig part detection and association with a fully-convolutional network.  Sensors, 19(4):852, 2019.
  * [23] Helena Russello, Rik van der Tol, and Gert Kootstra.  T-LEAP: Occlusion-robust pose estimation of walking cows using temporal information.  Computers and Electronics in Agriculture, 192:106559, 2022.
  * [24] Ke Sun, Bin Xiao, Dong Liu, and Jingdong Wang.  Deep high-resolution representation learning for human pose estimation.  In Proceedings of the IEEE/CVF conference on Computer Vision and Pattern Recognition, pages 5693–5703, 2019.
  * [25] Elsbeth A. van Dam, Lucas P.J.J. Noldus, and Marcel A.J. van Gerven.  Deep learning improves automated rodent behavior recognition within a specific experimental setup.  Journal of Neuroscience Methods, 332:108536, 2020.
  * [26] Paul Voigtlaender, Michael Krause, Aljosa Osep, Jonathon Luiten, Berin Balachandar Gnana Sekar, Andreas Geiger, and Bastian Leibe.  MOTS: Multi-Object Tracking and Segmentation.  arXiv:1902.03604[cs], 2019.  arXiv: 1902.03604.
  * [27] Fasheng Wang, Ping Cao, Fu Li, Xing Wang, Bing He, and Fuming Sun.  WATB: Wild Animal Tracking Benchmark.  International Journal of Computer Vision, 131(4):899–917, 2023.
  * [28] Zhi Weng, Zhengguang Mei, Zhiqiang Zheng, and Caili Gong.  Video-based multi-scene cow pose estimation.  In Proceedings of the 2023 3rd International Conference on Bioinformatics and Intelligent Computing, pages 187–194, 2023.
  * [29] Hang Yu, Yufei Xu, Jing Zhang, Wei Zhao, Ziyu Guan, and Dacheng Tao.  AP-10K: A Benchmark for Animal Pose Estimation in the Wild.  In Thirty-fifth Conference on Neural Information Processing Systems Datasets and Benchmarks Track (Round 2), 2021.
  * [30] Libo Zhang, Junyuan Gao, Zhen Xiao, and Heng Fan.  AnimalTrack: A Benchmark for Multi-Animal Tracking in the Wild.  International Journal of Computer Vision, 131(2):496–513, 2023.
  * [31] Yifu Zhang, Peize Sun, Yi Jiang, Dongdong Yu, Fucheng Weng, Zehuan Yuan, Ping Luo, Wenyu Liu, and Xinggang Wang.  ByteTrack: Multi-Object Tracking by Associating Every Detection Box.  In Proceedings of the European Conference on Computer Vision (ECCV), 2022.