# Splitting Groups with Basis Property

Authors: Abdullah Aljouiee
Journal: Chinese Journal of Mathematics
 (2014)

---

## Abstract

A finite groupG is called splitting or splittable if it is a union of some collections of its proper subgroups intersecting pairwise at the identity. A special kind of splitting is known to be normal splitting. Also, a group G is said to have the basis property if, for each subgroup H
≤
G, H has a basis (minimal generating set), and any two bases have the same cardinality. In this work, I discuss a relation between classes of finite groups that possess both normal splitting and the basis property. This paper shows mainly that any non-p-group with basis property is normal splitting. However, the converse is not true in general. A counterexample is given. It is well known that any p-group has basis property. I demonstrate some types of p-groups which are splitting as well.


---

## Body

## 1. Introduction

Miller [1] was the first one who studied the concept of splitting groups; then it was discussed by many authors such as Young [2] who gave a precise definition of this concept. Hugher and Thompson [3] introduced the definition of HT-groups as follows: a finite nilpotent group G (not p-group) is called HT-group if there exists a prime number p and a subgroup H which is generated by all elements of order not equal to p with H
≠
G and p
∣
|
H
|.In Baer [4, 5], Kegel [6], and Kontorovich [7], it was given a classification of solvable non-p-groups that are splitting. Also, Isaacs [8] studied the so-called equally splitting. Zassenhaus [9] had a great role in studying splitting groups, and Suzuki [10] had a crucial contribution in this regard by finding a new class of simple splitting groups, and finishing the topic of classifying unsolvable splitting finite groups. Sozutov and Shlepkin [11] generalized Kegel’s theorem [6] on finite HT-groups and Baer’s theorem [5]. I use the same notation and terminology that appeared in the references, especially [5, 12]. In [13], it is shown that any finite group with basis property is either elementary p-group, nonelementary p-group, or a group that can be written as P
⋊
H, where P is of order p
a and H is a cyclic q-group of order q
b where p, q are distinct primes and a
,
b
∈
ℤ
+. This work shows that all finite groups with basis property of the first or third kind are splitting, while groups of the second kind may or may not be splitting.LetG be a nontrivial finite group. A collection {
G
i
} of nontrivial subgroups of G is said to be splitting of G if every element g
≠
1 of G is contained in one and only one subgroup of {
G
i
}, and {
G
i
} is called nontrivial if G
i
≠
G for all i.LetG be a group with splitting {
G
i
}; then the subgroup G
i is called a component of this splitting. So, one has
(1)
G
i
∩
G
j
=
{
1
}
,
if
i
≠
j
.
Let H be a subgroup of G; then the collection of subgroups H
i of H defined by H
i
=
G
i
∩
H is a splitting of H. This is called the induced splitting. The induced splitting is nontrivial if and only if H is contained in no component of the splitting {
G
i
}.Lemma 1 (Young [2]).
If a torsion abelian group has a splitting, then there exists a prime numberp so that each nonidentity element is of order p. Moreover, if |
G
|
>
p (prime number), and the nonidentity elements of G have order p, then G has a nontrivial splitting.Lemma 2 (Miller [1]).
An abelian group of finite order has a nontrivial splitting if and only if it is of orderp
m (m
>
1), and it must be an elementary abelian p-group.


## 2. Some Results on Splittingp-Groups

In 1960-1961, Baer [4, 5], Kegel [6], and Suzuki [10] obtained the classification of finite splitting groups that are not p-groups, such groups must be either simple or solvable.Theorem 3 (the classification theorem).
LetG be a finite splitting group which is not p-group. Then G is isomorphic to one of the following groups:(i)
S
4
;
(ii)
P
G
L
(
2
,
q
)
where
q
≥
5
,
q
is odd;
(iii)
P
S
L
(
2
,
q
)
where
q
≥
4
;
(iv)
Suzuki simple group,
S
Z
(
2
2
n
+
1
)
,
n
∈
ℤ
+
;
(v)
Frobenius group;
(vi)
HT-group (the Hugher and Thompson group).Proposition 4.
Consider the HT-group, and keep the notation in its definition. Ifp
=
2, then H is abelian.Proof.
Right from the definition of HT-group,H is generated by all elements of order greater than 2. Let g
∈
G
∖
H, so g
2
=
1. For an arbitrary element h
∈
H, one has h
g
∉
H, since, otherwise, h
-
1
(
h
g
)
=
g would be in H, and this contradicts the assumption. Thus h
g is of order 2 or g
h
g
-
1
=
h
-
1. Now, for h
,
h
′
∈
H, we get h
-
1
h
′
-
1
=
g
h
g
-
1
g
h
′
g
-
1
=
g
h
h
′
g
-
1
=
(
h
h
′
)
-
1
=
h
′
-
1
h
-
1. Thus H is abelian.Proposition 5.
LetG be a nonelementary 2-group. Then G is splitting if and only if G
=
H
⋊
〈
b
〉 where H is abelian 2-group, containing an element of order 4, b
2
=
1 and b
h
b
-
1
=
h
-
1, ∀
h
∈
H.Proof.
Assume thatG
=
H
⋊
〈
b
〉; then G is a Frobenius group with abelian kernel, so by Theorem 3, G is splitting.
Conversely, assume thatG is splitting; then byKontorovich [7], all elements of G whose order is greater than 2 are contained in one component H of the splitting of G. If an element b
∈
G
∖
H, then for an arbitrary element h
∈
H, the element h
b
∉
H. Hence (
h
b
)
2
=
1. Since h
b
h
b
=
1, then b
h
b
=
h
-
1
,
∀
h
∈
H. Thus H is abelian as we have seen in Proposition 4. Since G is nonelementary 2-group, H must have an element x of order 2
k, k
≥
2. Thus the element x
2
k
-
2 is of order 4.
Finally, we will prove thatG
=
H
⋊
〈
b
〉. Suppose that c
∈
G
∖
H. Then, as mentioned before, c
h
c
-
1
=
h
-
1, ∀
h
∈
H. Hence, (
b
c
)
h
(
b
c
)
-
1
=
b
(
c
h
c
-
1
)
b
-
1
=
b
h
-
1
b
-
1
=
h, so, b
c commutes with every element in H. Since H contains elements with fourth order, then by Kontorovich [14], b
c
∈
H. Hence, c
∈
H
⋊
〈
b
〉 and G
=
H
⋊
〈
b
〉.A splitting is called abelian if it consists of two components, one of them (G
1 say) is abelian, containing all elements of nonprime order, and G
1 can be written as G
1
=
(
〈
a
1
〉
×
〈
a
2
〉
×
⋯
×
〈
a
p
-
1
〉
)
⋊
〈
b
〉, where a
1
p
2
=
a
2
p
=
a
3
p
=
⋯
=
a
p
-
1
p
=
b
p
=
1 and a
i
+
1
=
[
b
,
a
i
] for i
=
1,2
,
…
,
p
-
2, [
b
,
a
p
-
1
]
=
a
1
-
p.Proposition 6.
LetP be a p-group with minimal abelian splitting which contains elements of order p
2. Let N
⊲
P be an elementary abelian group such that [
P
:
N
]
=
p. Then, P is a direct product of dihedral group D
4 of order 8 and elementary 2-group. In particular p
=
2.Proof.
Assume thatP is not 2-group. Since P has abelian splitting and it contains a normal subgroup N of index p, then P
=
H
⋊
〈
b
〉 where H is an abelian component containing all elements of composite order. And all cyclic subgroups that are not contained in H form abelian splitting, see [15]. Hence, by [7], the center Z
(
P
) of P is contained in the abelian components of P which is generated by all elements of composite order. If h
∈
H is of order p
2, then we prove that Z
(
P
)
=
〈
h
p
〉. If x
∈
Z
(
P
)
∖
〈
h
〉, then P
/
〈
x
〉 is p-group with abelian splitting containing element of order p
2 and |
P
/
〈
x
〉
|
<
|
P
|. This leads to a contradiction. So, Z
(
P
)
≤
〈
h
〉. On the other hand, 〈
h
〉
≰
Z
(
P
) because, otherwise, 〈
h
,
b
〉 would be an abelian splitting of P. Thus Z
(
P
)
=
〈
h
p
〉. In this case, by [15], P corresponds to the Wreath product of two cyclic subgroups of P with prime orders. Hence, P is not splitting if p
≠
2. But this forces p to be equal to 2. This means that P
=
H
⋊
〈
b
〉 where b
2
=
1 and b
h
b
-
1
=
h
-
1, ∀
h
∈
H, and H is abelian 2-group, containing an element of order 4. So, by Proposition 5, P is 2-group and P
=
D
4
×
H
1 where D
4 is dihedral group of order 8 and H
1 is an elementary 2-group.


## 3. Normal Splitting

Let{
G
i
} be a splitting of G and {
K
s
} another splitting of G; then the collection of intersections G
i
∩
K
s is a splitting of G. A splitting {
K
s
} is said to be a refinement of another splitting {
G
i
} if every component K
s is contained in a component of G
i. Thus, a splitting {
K
s
} is a trivial refinement of {
G
i
}, if every K
s of the splitting {
K
s
} is either trivial group; that is, K
s is the identity or K
s
=
G
i for some i. Hence if the splitting {
G
i
} admits no nontrivial refinement, then it is called a minimal splitting.Definition 7.
A normal splitting{
G
i
} is a splitting in which every conjugate subgroup g
-
1
G
j
g of a component G
j is again a component of {
G
i
}.Remark 8.
The sum of subgroups, which are conjugate ofH, forms the following normal subgroup:
(2)
H
-
=
H
+
H
(
g
2
)
+
⋯
+
H
(
g
s
)
.
Frobenius theorem gives the sufficient conditions to decompose Frobenius group (Frobenius group is a group G which has a subgroup H, such that H
=
N
G
(
H
) and H
∩
H
(
g
)
=
{
1
} for every g
∈
G
∖
H) into a direct sum. Then by Frobenius theorem, G is decomposable into a direct sum of subgroups; that is, G
=
P
⊕
H
-, such that P is normal subgroup of G and
(3)
H
-
=
H
+
H
(
g
2
)
+
⋯
+
H
(
g
j
)
,
(
[
G
:
H
]
=
j
)
.Example 9.
The symmetric groupG
=
〈
{
σ
,
τ
∣
σ
2
=
τ
3
=
1
,
τ
σ
=
σ
τ
2
}
〉 is normal splitting group, since it is equal to 〈
σ
〉
∪
〈
σ
τ
〉
∪
〈
σ
τ
2
〉
∪
〈
τ
〉 and G
=
〈
τ
〉
⋊
〈
σ
〉. Take G
1
=
〈
σ
〉, and verify that 〈
σ
〉
∩
α
〈
σ
〉
α
-
1
=
{
1
} for all α
∈
G
∖
〈
σ
〉.Any splitting can be refined. A splitting is called minimal if it cannot be refined anymore. This has been shown in the following.Lemma 10 (see [10]).
Every splitting has a refinement which is normal.Proposition 11.
Any minimal splitting is normal.Proof.
It is straightforward that if{
G
i
} is splitting, then {
g
G
i
g
-
1
} is again splitting for any g
∈
G. Now consider the refinement splitting g
G
i
g
-
1
∩
G
j and notice that {
G
i
} is minimal. Then, g
G
i
g
-
1
∩
G
j is either 1 or G
j. In both cases G
j
⊆
g
G
i
g
-
1. Similarly one has G
s
⊆
g
G
j
g
-
1
⊆
G
j, so s
=
j and thus g
G
i
g
-
1
=
G
j. This shows that {
G
i
} is normal.Definition 12.
A finite groupG is said to have the basis property if any subgroup H
≤
G has a base (minimal generating set) and any two bases of H have the same cardinality.Definition 13.
A subgroupH of G is said to be isolated if for every cyclic subgroup C one has H
∩
C
≠
{
1
} implies that C
≤
H.In [12, 13], the sufficient and necessary conditions for a finite group to have the basis property were stated and proved. A special attention was given to groups that have abelian fitting subgroups. The following two theorems summarize such results.Theorem 14.
If a finite groupG possesses the basis property, then it is a p-group or G
=
P
⋊
〈
y
〉 where P is a p-group and 〈
y
〉 is a cyclic group of order q
b, p
≠
q are primes and b
∈
ℤ
+.I remind the reader with some basic definitions. The isotopic representation is the representation which can be written as a direct sum of equivalent irreducible representation.The Frattini subgroupΦ
(
P
) of a group P is the intersection of all maximal proper subgroups of P. Frattini of higher order is defined inductively as Φ
i
(
P
)
=
Φ
(
Φ
i
-
1
(
P
)
). The fitting subgroup of a finite group is the unique maximal nilpotent normal subgroup. So we have the following theorem.Theorem 15.
Let a finite groupG be a semidirect product of p-group P
=
F
i
t
(
G
) by a cyclic q-group 〈
y
〉 of order q
b, p
≠
q are primes and b
∈
ℤ
+. Then, the group G has the basis property if and only if for any element u
∈
〈
y
〉
∖
{
1
} and for any normal subgroup N
⊴
P, the induced automorphism φ
u must define an isotopic representation on every quotient Frattini subgroup.As in [13], finite groups with basis property are either p-groups which are elementary, that is, the order of any element is less than or equal to p, or p-groups which are not elementary, or of mixed order p
a
q
b where p
,
q are distinct primes and a
,
b
∈
ℤ
+.Theorem 16.
LetG be a finite non-p-group with basis property such that G
=
P
⋊
〈
y
〉, where P
⊴
G is the maximal nilpotent normal subgroup of G of order p
a
(
P
=
F
i
t
(
G
)
), and 〈
y
〉 is a cyclic q-group of order q
b where p, q are primes, p
≠
q, and a
,
b
∈
ℤ
+. Then G has a normal splitting.Proof.
SinceP
⊴
G, then P is a p-Sylow subgroup of G, which contains all p-elements of G. Therefore, the order of every element in G is either power of p or q. So, we know that by [13], p-elements do not commute with q-elements. Thus for every element x
∈
P, x
≠
1, we have C
G
(
x
)
⊆
P. C
G
(
x
) is the centralizer of x in G. Hence, P is isolated subgroup in G, so by [14], G is splitting group. That is, G
=
P
+
〈
y
-
〉 such that 〈
y
-
〉
=
〈
y
〉
+
〈
y
〉
(
x
1
)
+
⋯
+
〈
y
〉
(
x
r
) where [
G
:
〈
y
〉
]
=
r.
Now, we will show that〈
y
〉
∩
g
〈
y
〉
g
-
1
=
{
1
} for any g
∈
G
∖
〈
y
〉. If g
=
y
j
c for some c
∈
P
∖
{
1
} and 1
≤
j
≤
q
b, then g
-
1
y
g
=
c
-
1
y
-
j
y
y
j
c
=
c
-
1
y
c. So, we may assume that g
∈
P
∖
{
1
}. If t
∈
c
-
1
〈
y
〉
c
∩
〈
y
〉 where t
≠
1, one can find 1
≤
k, l
≤
q
b such that t
=
y
l
=
c
-
1
y
k
c or
(4)
y
-
k
c
y
l
=
c
.
Sincec
∈
P
∖
{
1
}, there is a number 0
≤
i
≤
j
P with c
∈
Φ
i
(
P
)
∖
Φ
i
+
1
(
P
). If we apply the homomorphism ϕ
:
G
→
G
╱
P on (4), we get 1
=
ϕ
(
c
)
=
ϕ
(
y
)
-
k
ϕ
(
c
)
ϕ
(
y
)
l
=
ϕ
(
y
)
l
-
k. Thus, q
b
∣
(
l
-
k
) or l
=
k, which leads to c
y
l
=
c
y
l. This later result, given that G possesses the basis property, contradicts, according to [13], the fact that p-elements do not commute with q-elements. So, c must be the identity. This means that G has a normal splitting.For the sake of the converse of Theorem16, one notices that in Theorem 3, S
4 does not have the basis property and cases (ii), (iii), and (iv) are simple, so they do not have basis property. The class of HT-group is nilpotent but the only nilpotent groups with the basis property are p-groups. The only possible true converse is Frobenius groups. In fact, this is not true in general. Here is a counterexample.Example 17.
The groupG
=
Z
19
4
⋊
Q
8, where Q
8 is the quaternion with 8 elements, is normal splitting group that does not have the basis property. We demonstrate this situation.
The quaternionQ
8
=
{
x
,
y
∣
x
4
=
x
y
x
y
=
1
,
x
2
=
y
2
}, and the semidirect product G
=
Z
19
4
⋊
Q
8 is defined by an automorphism λ
:
Q
8
→
Aut
(
Z
19
4
), which gives rise to a corresponding linear matrix representation for some base on the field GF
(
19
). Such a representation can be given in terms of
(5)
A
x
=
(
(
0
-
1
1
0
)
(
0
0
0
0
)
(
0
0
0
0
)
(
0
-
1
1
0
)
)
,
A
y
=
(
(
0
0
0
0
)
(
1
0
0
-
1
)
(
-
1
0
0
1
)
(
0
0
0
0
)
)
by considering Z
19
4 as a vector space on GF
(
19
). If ϕ
x
,
ϕ
y are operators represented by A
x, A
y for some base, respectively, then we get a linear representation ϕ for Q
8 whose kernel is 1. And we get that
(6)
ϕ
(
1
)
=
(
(
1
0
0
1
)
(
0
0
0
0
)
(
0
0
0
0
)
(
1
0
0
1
)
)
,
ϕ
(
x
2
)
=
(
(
-
1
0
0
-
1
)
(
0
0
0
0
)
(
0
0
0
0
)
(
-
1
0
0
-
1
)
)
,
ϕ
(
y
3
)
=
(
(
0
0
0
0
)
(
-
1
0
0
1
)
(
1
0
0
-
1
)
(
0
0
0
0
)
)
,
ϕ
(
x
3
)
=
(
(
0
1
-
1
0
)
(
0
0
0
0
)
(
0
0
0
0
)
(
0
1
-
1
0
)
)
,
ϕ
(
x
y
)
=
(
(
0
0
0
0
)
(
0
-
1
-
1
0
)
(
0
1
1
0
)
(
0
0
0
0
)
)
,
ϕ
(
y
x
)
=
(
(
0
0
0
0
)
(
0
1
1
0
)
(
0
-
1
-
1
0
)
(
0
0
0
0
)
)
.
So, we get all, linear representations for the elements of Q
8 which induce nonidentical minimal polynomials. So, we get a nonequivalent irreducible representation, and thus we do not have isotopic representation, but this violates one of the necessary conditions for finite groups with basis property as in the proof of Theorem 15 (see [13]). Thus Z
19
4
⋊
Q
8 does not have the basis property. From the other side, we show that this is a Frobenius group and so, it is a normal splitting group. Consider the subgroup H
=
〈
x
,
y
〉, and let u
∈
G
∖
H. If v
∈
H
∩
u
-
1
H
u, v
≠
1, then we have
(7)
u
=
h
c
,
where c
∈
Z
19
4 and h
∈
H. So, v
=
u
-
1
s
u, v
,
s
∈
H. That is to say
(8)
v
=
c
-
1
h
-
1
s
h
c
.
Applying the homomorphism θ
:
G
→
G
/
Z
19
4 on (7) and (8) leads to θ
(
h
c
)
=
h for all h
∈
H, c
∈
Z
19
4, and hence v
=
θ
(
v
)
=
h
-
1
s
h. This implies that h
-
1
s
h
=
u
-
1
s
u or u
h
-
1 commutes with s. That is u
h
-
1
∈
H or u
∈
H which contradicts the assumption. Henceforth, H
∩
u
-
1
H
u
=
{
1
} and G is a Frobenius group with abelian kernel.



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*Source: 950572-2014-02-06.xml*