arXiv:1803.02420v1 [] 6 Mar 2018

CLASSIFICATION OF GROUPS ACCORDING TO THE
NUMBER OF END VERTICES IN THE COPRIME GRAPH

TARIQ A. ALRAQAD
Department of Mathematics, University of Hail
Hail, Kingdom of Saudi Arabia
t.alraqad@uoh.edu.sa
MUHAMMAD S. SAEED
Department of Mathematics, University of Hail
Kingdom of Saudi Arabia
ms.saeed@uoh.edu.sa
ETAF S. ALSHAWARBEH
Department of Mathematics, University of Hail
Kingdom of Saudi Arabia
e.alshawarbeh@uoh.edu.sa
Abstract. In this paper we characterize groups according to the number of
end vertices in the associated coprime graphs. An upper bound on the order
of the group that depends on the number of end vertices is obtained. We
also prove that 2−groups are the only groups whose coprime graphs have odd
number of end vertices. Classifications of groups with small number of end
vertices in the coprime graphs are given. One of the results shows that Z4
and Z2 × Z2 are the only groups whose coprime graph has exactly three end
vertices.
Keywords: Coprime Graphs, Finite Groups, End Vertices.
Mathematics Subject Classification: Primary 20F65, Secondary 05C25

1. Introduction
Throughout this paper, G denotes a finite group. The order of a group G is
denoted by |G| and the order of an element g ∈ G is denoted by |g|. For a general
reference on group theory and graph theory, we refer the reader to the text-books
of [7] and [1] respectively. The greatest common divisor of two integers a and b
is denoted by (a, b). The radical of an integer n ≥ 2, denoted by rad(n), is the
product of the distinct prime divisors of n, and rad(1) = 1.
The coprime graph of a finite group G is the simple graph ΓG whose vertices
are the elements of G with two vertices x and y are adjacent if (|x|, |y|) = 1. It is
clear that the identity element of G (usually denoted by e) is adjacent with every
other element; so the graph is connected with diameter at most 2. The concept of
coprime graphs was introduced by Sattanathan and Kala [8] where they call them
order prime graphs of finite groups. Later Ma et al [5] studied these graphs and
called them coprime graphs of groups. Further studies of these graphs were done
in [3, 6, 9].
The degree of a vertex in a graph is the number of vertices in the graph that are
adjacent to it. A vertex that has degree one is called an end vertex. For a group G,
we denot by EG the set of all end vertices in ΓG other than the identity element.
1

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Note that EG may be empty. In [5, section 3], Ma et al characterized the groups
whose coprime graphs have exactly 0, 1, or 2 end vertices, that is |EG | = 0, 1, 2.
Then they ended the section with the following question:
Question 1. [5, Question 3.7] Is it possible to characterize all finite groups G whose
coprime graph contains precisely three end vertices?
In [3], Dorbidi has proved a result that gives the number of end vertices in a
coprime graph of any nilpotent group.
In this paper our gaol is to classify groups according to the number of the end
vertices in the associated coprime graphs. In Section (2), we present the main
results. We prove that one of the end vertices of ΓG has order equal to a power of
a prime p if and only if G is a p−group. We conclude from this result that |EG |
is odd if and only if G is a 2−group. In addition, we obtain an upper bound on
the order of the group that depends on the size of EG . Section (3) is devoted to
classifications of groups whose coprime graphs have small number of end vertices
mainly between 1 and 10. Corollary (2) states that Z4 and Z2 × Z2 are the only
groups whose coprime graph has exactly three end vertices. This gives a complete
answer to [5, Question 3.7].
2. Main Result
We start this section by the following result, which characterize the coprime
graphs of p−groups. For proof see [8, Theorem 2.7] or [5, Proposition 2.6]
Theorem 1. Let G be a group. Then ΓG ∼
= K1,|G|−1 (star graph) if and only if G
is a p−group with some prime integer p.
Now we prove the following useful lemma.
Lemma 1. Let G be a group with |G| ≥ 3. Then
(i) EG = {x ∈ G | rad(|x|) = rad(|G|)},
(ii) for each x ∈ EG , φ(|x|) ≤ |EG |, (φ denotes the Euler’s phi function) and
(iii) if there exists x ∈ EG such that φ(|x|) = |EG |, then rad(|x|) = |x| and hxi is
a unique cyclic subgroup of G of order |x|.
Proof. (i) By Cauchy Theorem we note that for each prime number p dividing |G|
there is an element of G whose order is p. So an element x ∈ G is an end vertex of
ΓG if and only if every prime divisor of |G| divides |x|, which is equivalent to say
that rad(|x|) = rad(|G|).
(ii) Let x ∈ EG . Then any generator of hxi is in EG . Since hxi has exactly φ(|x|)
generators, we must have φ(|x|) ≤ |EG |.
(iii) Suppose that φ(|x|) = |EG |. Then EG consists of the generators of hxi, that
|x|
. Then
is EG = {xj | (j, |x|) = 1}. Assume that rad(|x|) < |x| and let d = rad(|x|)
d
d
d
|xd | = |x|
d = rad(|x|). So rad(|x |) = rad(|x|), which yields x ∈ EG . But x is
not a generator of hxi, a contradiction. So rad(|x|) = |x|. Moreover, if |y| = |x| for
some y ∈ G, then y ∈ EG . So hyi = hxi. This completes the proof.


Theorem 2. Let G be a finite group. Then EG contains a non-identity element
whose order is a power of prime p if and only if G is a p−group. Moreover, |G| =
|EG | + 1.

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Proof. Let x be an end vertex in ΓG such that |x| = pk for some positive integer
k. Then by Lemma (1), p = rad(|x|) = rad(|G|), and hence G is a p−group.
Moreover, Theorem (1) implies that ΓG is a star graph, which yields every nonidentity element of G is an end vertex. Hence |G| = |EG | + 1. The converse follows
directly by applying Theorem (1).

The following result gives a characterization of all finite groups whose coprime
graphs have odd number of end vertices.
Theorem 3. Let G be a finite group. Then |EG | is odd if and only if G is a
2−group. Moreover, |EG | = 2n − 1 for some positive integer n.
Proof. Suppose |EG | is odd. Since |x| = |x−1 | for every x ∈ G, x ∈ EG if and
only if x−1 ∈ EG . Furthermore, |EG | is odd, so there must be x ∈ EG such that
x = x−1 . This implies that EG contains an element of order 2. Thus by Theorem
(2) we conclude that G is a 2−group, and |EG | = |G| − 1.
Now, if G is a 2−group, then |EG | = |G| − 1 which is odd. This proves the
converse.

In order to classify all groups (up to isomorphism) whose coprime graphs have
the same number of end vertices, a natural question arises is that how many possible
orders of such groups are there? In the case |EG | = 0, one can find that there are
infinitely many groups whose coprime graphs have no end vertices. For instance,
if q ≥ 5 is a prime then ES3 ×Zq is empty. As a matter of fact if G is a nonabelian
finite group and rad(|G|) = |G| then EG is empty. For the remainder of this section
the goal is to establish an upper bound on the order of the group G depending on
the size of EG . We will only be concerned with the cases when |EG | is even because
the cases when |EG | is odd are completely solved in Theorem (3).
The centralizer of an element x in a group G is defined to be CG (x) = {g ∈ G |
gx = xg}, and the conjugacy class of G containing x is ClG (x) = {gxg −1 | g ∈ G}.
It is well known in group theory that CG (x) is a subgroup of G and its index equal
to the size of ClG (x); that is [G : CG (x)] = |ClG (x)|. The following theorem gives
a characterization of the centralizers of the end vertices.
Theorem 4. Let G be a group and let x ∈ EG . Then
[
CG (x) =
hyi .
y∈EG ∩CG (x)

S

Proof. Let H = y∈EG ∩CG (x) hyi. If y ∈ EG ∩ CG (x) then y ∈ CG (x), and hence
hyi ⊆ CG (x). So H ⊆ CG (x). Now we need to show that CG (x) ⊆ H. Clearly
EG ∩CG (x) ⊆ H. Now, let g ∈ CG (x). If rad(|g|) = rad(|G|) then g ∈ EG ∩CG (x) ⊆
F . Assume rad(|g|) < rad(|G|), then we need to show that g ∈ hyi for some
y ∈ EG ∩ CG (x). Let d = rad(|G|)/rad(|g|) and let z = x|x|/d . Then we have
(|z|, |g|) = (d, |g|) = 1 and zg = gz. So |gz| = |g||z|, which implies rad(|gz|) =
rad(|g|)rad(|z|) = rad(|G|). So gz ∈ EG ∩ CG (x), which yields hgzi ⊆ F . Now,
d

(gz)d = gx|x|/d = g d x|x| = g d .
Also since (|g|, d) = 1, we have hgi = g d . So hgi = (gz)d , and hence g ∈

(gz)d ⊆ F . Therefore CG (x) ⊆ F .
The next theorem gives an upper bound on the order of G depending on |EG |.

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Theorem 5. Let n be a positive integer and M be the largest possible integer such
that φ(M ) ≤ 2n. If |EG | = 2n then
|G| ≤ 2n(M n − n + 1).
.
Proof. Assume that |EG | = 2n and let x ∈ EG . Then ClG (x) ⊆ EG , and so
|ClG (x)| ≤ |EG | = 2n. Let m = min φ(|y|). Then for each y ∈ EG ∩ CG (x), the
y∈EG

size of the set {z ∈ EG ∩ CG (x)| hzi = hyi} is at least m. Also we know that e ∈ hyi
for all y. Theorem (4) implies that
(2.1)

|CG (x)| =

[

hyi ≤ 1 +

y∈EG ∩CG (x)

1
m

X

(|y| − 1).

y∈EG ∩CG (x)

Now, from Lemma (1) it is clear that |y| ≤ M for all y ∈ EG . Substituting this
in 2.1 yields
|CG (x)| ≤ 1 +

(M − 1)|EG ∩ CG (x)|
.
m

Since no element in EG has order 2, we obtain that m ≥ 2. Thus

|CG (x)| ≤ 1+

(M − 1)|EG ∩ CG (x)|
(M − 1)|EG |
(M − 1)2n
≤ 1+
= 1+
= M n−n+1.
2
2
2

|G| = [G : CG (x)]|CG (x)| = |ClG (x)||CG (x)| ≤ 2n(M n − n + 1).

Under some conditions the bound given in Theorem (5) can be improved. The
following theorem gives a better bound on the size of the group under the condition
that all end vertices are generated by the same element. We find this results is very
useful in the next section.
Theorem 6. Let G be a group with |G| ≥ 3 and |EG | ≥ 1. If there exists x ∈ G
such that EG ⊆ hxi, then |G| ≤ |x|φ(|x|).
Proof. Assume that EG ⊆ hxi for some x ∈ G and let xt ∈ EG . So rad(|xt |) =
rad(|G|). Since rad(|xt |) divides rad(|x|) which divides rad(|G|), we obtian that
rad(|x|) = rad(|G|), which yields x ∈ EG . This implies that ClG (x) ⊆ EG ⊆ hxi.
Thus |ClG (x)| ≤ φ(|x|). Also Theorem (4) implies that CG (x) ⊆ hxi. Thus we have
[G : hxi] = [G : CG (x)] = |Cl (x)| ≤ φ(|x|).
Hence |G| = [G : hxi]|x| ≤ |x|φ(|x|).



Remark 1. Taking |EG | = n in Theorem (6) yields |G| ≤ M n. This bound is met
by some groups as we will see later in Theorem (7).

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3. Classifications of groups with small number of end vertices
In this section we use the theorems presented in Section (2) to classify groups
whose coprime graphs have small number of end vertices. We start with the odd
cases because they follow directly from Theorem (3). It is shown in [5] that |EG | = 1
if and only if G ∼
= Z2 . We list this result here for completeness and we give a shorter
proof for it.
Corollary 1. Let G be a finite group. Then |EG | = 1 if and only if G ∼
= Z2 .
Proof. |EG | = 1 ⇔ |G| = 2 ⇔ G ∼
= Z2 .



The following result gives a complete answer to question 3.7. in [5].
Corollary 2. For a finite group G, |EG | = 3 if and only if G ∼
= Z2 × Z2 .
= Z4 or G ∼
Proof. |EG | = 3 ⇔ |G| = 4 ⇔ G ∼
= Z4 or G ∼
= Z2 × Z2 .



Theorem (3) states that the only possible odd values for |EG | are of the form
2n − 1. So there are no groups G for which EG consists of five elements, nine
elements, eleven elements, ... etc. Using similar arguments as in Corollaries (1)
and (2) it can be shown that |EG | = 7 if and only if |G| = 8, which means G could
be any one of the five groups of order eight.
For an integers k ≥ 2, D2k denote the dihedral group of order 2k, and Dic4k
denote the dicyclic group of order 4k which has presentation ha, b | ak = b2 =
e, bab−1 = a−1 i. For a prime power q, GA(1, q) denotes the general affine group of
degree 1 over the field of q elements. Next we deal with some cases in which |EG |
is even. The following theorem classifies all groups whose coprime graphs have two
end vertices, and this is a complete improvement of [5, Theorem 3.5].
Theorem 7. For a finite group G, |EG | = 2 if and only if G ∼
= Z3 , Z6 , D12 , Dic12
Proof. Suppose |EG | = 2. If there is x ∈ EG such that rad(|x|) = p, where p is a
prime, then G is a p−group and |G| = |EG | + 1 = 3. Thus G ∼
= Z3 .
Now assume that rad(|x|) is not a prime for all x ∈ EG . Form Lemma (1), we
have that φ(|x|) ≤ |EG | = 2 for all x ∈ EG , which yields |x| = 6 for all x ∈ EG .
Since an element and its inverse have the same order, there exists x ∈ G such that
|x| = 6 and EG = {x, x5 } ⊆ hxi. Thus Theorem (6) implies that |G| ≤ |x|φ(|x|) =
12. Furthermore, rad(|G|) = rad(|x|) = 6. So |G| = 6 or |G| = 12. If |G| = 6, then
G = hxi ∼
= Z6 . Also, from the groups of order 12 only D12 and Dic12 has exactly
two end vertices. Thus G ∼

= Z6 , D12 , Dic12 .
Remark 2. Each one of the groups D12 and Dic12 meet the bound given in Remark
(1). Each group contains element of order 6 that generate both end vertices and
2 × 6 = 12 which is the order of each group.
Theorem 8. For a finite group G, |EG | = 4 if and only if G ∼
= Z5 , Z10 , D20 ,
Dic20 ,GA(1, 5) × Z2 , or ha, b | a5 = b8 = e, bab−1 = a2 i.
Proof. Suppose |EG | = 4. If there is x ∈ EG such that rad(|x|) = p, where p is a
prime, then G is a p−group and |G| = |EG | + 1 = 5. So G ∼
= Z5 .
Now assume that rad(|x|) is not a prime for all x ∈ EG . By Lemma (1), we
have φ(|x|) ≤ 4. So the possible orders of the elements in EG are 6, 10, and 12. If
|x| = 12 for some x ∈ EG , then |x| = |x5 | = |x7 | = |x11 | = 12 and |x2 | = |x10 | = 6.
This implies that |EG | ≥ 6, a contradiction.

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Now Assume that |x| = 10 for some x ∈ EG . Then rad(|G|) = 10 and EG =
{x, x3 , x7 , x9 } ⊆ hxi. Now Theorem 6 implies that |G| ≤ |x|φ(|x|) = 40. So
|G| = 10, 20, or 40. After examining all groups of orders 10, 20, and 40 we conclude
that G ∼
= Z10 , D20 , Dic20 , GA(1, 5) × Z2 , or ha, b | a5 = b8 = e, bab−1 = a2 i.
The remaining case is that |x| = 6 for all x ∈ EG . In this case EG = {x, x5 , y, y 5 }
where x, y ∈ G with x 6= y, x 6= y −1 , and |x| = |y| = 6. Now yxy −1 ∈ EG but also
yxy −1 6= y and yxy −1 6= y −1 . So either yxy −1 = x or yxy −1 = x5 .
Case 1: yxy −1 = x. Then Theorem (4) implies that CG (x) = hxi ∪ hyi. So
|CG (x)| ≤ 11, which yields |CG (x)| = 6. So CG (x) = hxi, which implies y ∈ hxi, a
contradiction because y 6= x and y 6= x−1 .
Case 2: yxy −1 = x−1 . So y, y −1 ∈
/ CG (x). Then again Theorem (4) implies
CG (x) = hxi. Thus we have
|G| = [G : CG (x)]|CG (x)| = |ClG (x)||x| ≤ 24
So |G| = 6, 12, 18, or 24. Examining all groups of these orders shows that none of
them has exactly four elements of order 6.

Theorem 9. For a finite group G, |EG | = 6 if and only if G is isomorphic to one
of the following groups:
(i) (Groups of order 7) Z7 .
(ii) (Groups of order 12) Z12 , Z2 × Z6 .
(iii) (Groups of order 14) Z14 .
(iv) (Groups of order 18) S3 × Z3 .
(v) (Groups of order 24) D24 , Dic24 , S3 × Z4 , Dic12 × Z2 , D12 × Z2 , ha, b | a3 =
b8 = e, bab−1 = a−1 i, ha, b, c | a2 = b2 = c3 = (ac)2 = (ba)4 = e, bc = cbi.
(vi) (Groups of order 28) D28 , Dic28 .
(vii) (Groups of order 36) C3 × A4 , a, b | a9 = b2 = e, (a−1 b)2 = ba−2 .
(viii) (Groups of order 72)
ha, b, c | a2 = b2 = c9 = (ac)2 = e, bac−1 = cab, abcb = bac, bc3 = c3 bi,
ha, b, c, d | a2 = b2 = c3 = d3 = (bc)3 = (abc)2 = e, ad = d2 a, ac = c2 a, cd =
dc, bd = dbi.
Proof. Suppose |EG | = 6. Assume EG contains an element whose order is a power
of a prime p, then G is a p− group and |G| = |EG | + 1 = 7. So G ∼
= Z7 .
Now assume the order of any element of EG is not a power of a prime. By Lemma
(1), we have φ(|x|) ≤ 6. So the possible orders of the elements in EG are 6, 10, 12,
14, and 18. If |x| = 18 for some x ∈ EG , then |x| = |x5 | = |x7 | = |x11 | = |X 13 | =
|X 17 | = 18 and |x3 | = |x15 | = 6. This implies that |EG | ≥ 8, a contradiction.
Case 1: |x| = 14 for some x ∈ EG . Then rad(|G|) = 14 and
EG = {x, x3 , x5 , x9 , x11 , x13 } ⊆ hxi .
Now Theorem (6) implies that |G| ≤ |x|φ(|x|) = 84. So |G| = 14, 28, or 56.
Case 2: |x| = 12 for some x ∈ EG . Then rad(|G|) = 6 and
EG = {x, x2 , x5 , x7 , x10 , x11 } ⊆ hxi .
Again Theorem 6 implies that |G| ≤ |x|φ(|x|) = 48. So |G| = 12, 24, 36, or 48.
Case 3: |x| = 10 for some EG . Then the set U = {x, x3 , x7 , x9 } ⊆ EG }. So there
is y ∈ (EG \ U ) such that φ(|y|) = 2 and rad(|y|) = 10, which is impossible.
Case 4: |x| = 6 for all x ∈ EG . In this case EG = {x, x5 , y, y 5 , z, z 5 }, where
x, y, and z are elements of order 6 order in G. Also we have rad(|G| = 6) and

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|ClG (x)| ≤ 6. Furthermore, Theorem (4) implies that |CG (x)| ≤ 16. Hence |G| =
6, 12, 18, 24, 36, 48, 54, 72, 96.
From all cases combined we conclude that the possible orders for G are 6, 14, 12,
18, 24, 28, 36, 48, 54, 56, 72, 96. After analyzing the all coprime graphs of groups
having these orders we obtain the result in theorem.

For any positive integer n, the groups G, for which |EG | = n, can be determined
using the same technique presented in this section. We classified all groups G
for which |EG | = 8 or |EG | = 10. We mention these results for completeness
however the proofs are omitted. Regarding the case |EG | = 8, there are 48 groups
G for which |EG | = 8. We list below the GAP ID’s [4] for these groups. These
groups can be obtained from the GAP’s SmallGroup library using the command
”SmallGroup(n, k)” , where (n, k) is the ID of the group.
(9, 1), (9, 2), (15, 1), (18, 2), (18, 5), (24, 3), (24, 13), (30, 4), (36, 1), (36, 4), (36, 7),
(36, 13), (48, 28), (48, 29), (48, 30), (48, 48), (60, 1), (60, 2), (60, 3), (60, 10), (60, 11),
(60, 12), (72, 19), (72, 45), (120, 6), (120, 7), (120, 8), (120, 9), (120, 10), (120, 11),
(120, 12), (120, 13), (120, 14), (120, 40), (120, 41), (120, 42), (144, 114), (144, 120),
(144, 185), (144, 187), (240, 95), (240, 96), (240, 97), (240, 98), (240, 99), (240, 100),
(240, 101), (240, 195).
For the case |EG | = 10, we find that the groups Z11 , Z22 , D44 , and Dic44 are the
only groups whose coprime graph has exactly 10 end vertices.
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