Next Article in Journal
Extraction of Cellulose Nano-Whiskers Using Ionic Liquid-Assisted Ultra-Sonication: Optimization and Mathematical Modelling Using Box–Behnken Design
Previous Article in Journal
Implementation of Block-Based Hierarchical Prediction for Developing an Error-Propagation-Free Reversible Data Hiding Scheme

Symmetry 2019, 11(9), 1147; https://doi.org/10.3390/sym11091147

Article
On Finite Quasi-Core-p p-Groups
by 1,* and 2
1
Basic Course Department, Tianjin Sino-German University of Applied Sciences, Tianjin 300350, China
2
Department of Mathematics, Shanghai University, Shanghai 200444, China
*
Author to whom correspondence should be addressed.
Received: 4 August 2019 / Accepted: 5 September 2019 / Published: 10 September 2019

Abstract

:
Given a positive integer n, a finite group G is called quasi-core-n if x / x G has order at most n for any element x in G, where x G is the normal core of x in G. In this paper, we investigate the structure of finite quasi-core-p p-groups. We prove that if the nilpotency class of a quasi-core-p p-group is p + m , then the exponent of its commutator subgroup cannot exceed p m + 1 , where p is an odd prime and m is non-negative. If p = 3 , we prove that every quasi-core-3 3-group has nilpotency class at most 5 and its commutator subgroup is of exponent at most 9. We also show that the Frattini subgroup of a quasi-core-2 2-group is abelian.
Keywords:
finite p-group; quasi-core-p p-group; commutator subgroup

1. Introduction

Let G be a group and H is a subgroup of G. Then H G is the normal core of H in G, where H G = g G g 1 H g is the largest normal subgroup of G contained in H. A group G is called core-n if | H / H G | n for every subgroup H of G, where n is a positive integer. Buckley, Lennox, Neumaan, Smith and Wiegold investigated the core-n groups in [1]. They show that every locally finite group G with H / H G finite for all subgroups H is core-n for some n. Moreover, G has an abelian normal subgroup of index bounded in terms of n only. In [2], Lennox, Smith and Wiegold show that, for p 2 , a core-p p-group is nilpotent of class at most 3 and has an abelian normal subgroup of index at most p 5 . Furthermore, Cutolo, Khukhro, Lennox, Wiegold, Rinauro and Smith [3] prove that a core-p p-group G has a normal abelian subgroup whose index in G is at most p 2 if p 2 . Furthermore, if p = 2 , Cutolo, Smith and Wiegold [4] prove that every core-2 2-group has an abelian subgroup of index at most 16. As a deepening of research in this area, it is interesting to study the following question.
How about the structure of a p-group G in which | x / x G | p , for any x G ?
In this paper we hope to investigate the structure of a p-group G in which | x / x G | p , for any x G . For convenience, we call this kind of p-groups quasi-core-p p-groups.

2. Preliminaries

For convenience, we first recall some notations.
Let G be a p-group. We use d ( G ) and c ( G ) to denote the minimal number of generators and the nilpotency class of G respectively. We use C p m to denote the cyclic group of order p m . Let G n = [ g 1 , g 2 , , g n ] | g i G . If H and K are groups, then H × K denotes a product of H and K. For other notations the reader is referred to [5].
Lemma 1.
([6], Section Appendix 1, Theorem A.1.4) Let G be a p-group and x , y G .
1.
( x y ) p x p y p (mod 1 ( G ) G p ).
2.
[ x p , y ] [ x , y ] p (mod 1 ( N ) N p ), where N = x , [ x , y ] .
Lemma 2.
([7], Lemma 2.2) Suppose that G is a finite non-abelian p-group. Then the following conditions are equivalent.
1.
G is minimal non-abelian;
2.
d ( G ) = 2 and | G | = p ;
3.
d ( G ) = 2 and Φ ( G ) = Z ( G ) .
Lemma 3.
([8], Theorem) Let p be a prime and d , e positive integers. A regular d-generator metabelian p-group G whose commutator subgroup has exponent p e has nilpotency class at most e ( p 2 ) + 1 unless e = 1 , d > 2 , p > 2 when the class can be p. These bounds are best possible.
Lemma 4.
([9], Theorem 2) Let G be a metacyclic 2-group. Then G has one presentation of the following three kinds:
1.
G has a cyclic maximal subgroup.
2.
Ordinary metacyclic 2-groups G = a , b | a 2 r + s + u = 1 , b 2 r + s + t = a 2 r + s , a b = a 1 + 2 r , where r , s , t , u are non-negative integers with r 2 and u r .
3.
Exceptional metacyclic 2-groups G = a , b | a 2 r + s + v + t + u = 1 , b 2 r + s + t = a 2 r + s + v + t , a b = a 1 + 2 r + v , where r , s , v , t , t , u are non-negative integers with r 2 , t r , u 1 , t t = s v = t v = 0 , and if t r 1 , then u = 0 .
Groups of different types or of the same type but with different values of parameters are not isomorphic to each other.
Lemma 5.
([5], Theorem 10.3) Let G be a regular 3-group. Then G is abelian.
Lemma 6.
Let G be a quasi-core-p p-group. If H is a subgroup of G and N is a normal subgroup of G, then H and G / N are quasi-core-pp-groups.
Proof. 
The proof of the lemma comes immediately from the definition of quasi-core-p p-groups. □
Lemma 7.
Let G be a p-group. Then G is quasi-core-p if and only if x p G , for any element x in G.
Proof. 
Obviously, G is quasi-core-p if and only if | x G / x p | p , for any x G , and this holds if and only if x p G , for any element x in G. □
Lemma 8.
Let G be a quasi-core-p p-group. Then [ G , 1 ( G ) ] = 1 .
Proof. 
For any x G , according to Lemma 7, we see x p G . Thus G / C G ( x p ) is abelian and so G C G ( x p ) , which implies [ G , 1 ( G ) ] = 1 . □

3. Quasi-Core-p p-Groups with p > 2

In this section we investigate the quasi-core-p p-groups for p > 2 .
Theorem 1.
Let G be a quasi-core-p p-group and p > 2 . If G is cyclic, then | G | p .
Proof. 
Suppose the result is not true and G is a counterexample of minimal order. Then there exist a , b G such that o ( [ a , b ] ) p 2 . Thus we may assume G = a , b , [ a , b ] = c and L = a , c . Since G is regular, we may assume a b = 1 . By Lemma 1, we see [ a p , b ] = c p x , where x 1 ( L ) L p . Since L < G , 1 ( L ) L p = 1 . So x = 1 and [ a p , b ] = c p . Similarly, [ a , b p ] = c p . It follows from Lemma 7 that c p a b = 1 , in contradiction to the hypothesis. Thus the theorem is true. □
Corollary 1.
Let G be a quasi-core-p p-group with p > 2 . Then 1 ( G ) is abelian and 2 ( G ) Z ( G ) .
Proof. 
For any a , b G , we assume H = a p , b . By the hypotheses, we see a p G and so H is metacyclic. By Theorem 1, | H | p and so H is abelian or minimal non-abelian. Thus 1 ( H ) Φ ( H ) Z ( H ) by Lemma 2. It follows that [ a p 2 , b ] = [ a p , b p ] = 1 , which implies 1 ( G ) is abelian and 2 ( G ) Z ( G ) . □
Corollary 2.
Let G be a quasi-core-p p-group with p > 2 . Then G / C G ( a p ) C p , for any a G .
Proof. 
We may assume a p Z ( G ) and o ( a ) = p n . Then n 3 and there exists an element b G such that b C G ( a p ) . By Theorem 1, we may assume [ a p , b ] = a p n 1 . Take x G \ C G ( a p ) . Assume [ a p , x ] = a i p n 1 , where ( i , p ) = 1 . Then [ a p , b i x ] = 1 , which implies x C G ( a p ) b and so G = C G ( a p ) b . It follows from b p C G ( a p ) that G / C G ( a p ) C p . □
Corollary 3.
Let G be a quasi-core-p p-group with p > 2 . If c ( G / 1 ( G ) ) n , then c ( G ) n + 2 .
Proof. 
Set G ¯ = G / 1 ( G ) . Then G ¯ n + 1 = 1 ¯ and so G n + 1 1 ( G ) . It follows from Theorem 1 that [ G n + 1 , G ] [ 1 ( G ) , G ] Z ( G ) , which implies c ( G ) n + 2 . □
According to Lemma 3 and Corollary 3, we get the following theorem.
Theorem 2.
Suppose that G is a quasi-core-p p-group and G is abelian with p > 2 . If d ( G ) = 2 , then c ( G ) p + 1 . If d ( G ) > 2 , then c ( G ) p + 2 .
If p = 3 , then, according to Lemma 5 and Corollary 3, we get the theorem below.
Theorem 3.
Let G be a quasi-core-3 3-group. If d ( G ) = 2 , then c ( G ) 4 . If d ( G ) > 2 , then c ( G ) 5 .
Theorem 4.
Let G be a quasi-core-3 3-group with d ( G ) = 2 . Then Φ ( G ) is abelian.
Proof. 
We may assume G = x , y and [ x , y ] = z . Then G = z , [ z , g ] | g G . For any g 1 , g 2 G , it follows from Theorem 3 that [ z , [ z , g ] ] [ G 2 , G 3 ] = 1 and [ [ z , g 1 ] , [ z , g 2 ] ] = 1 , which implies G is abelian. So, according to Lemma 8 and Corollary 1, Φ ( G ) is abelian. □
Now, we investigate the exponent of commutator subgroups of the quasi-core-p p-groups.
Lemma 9.
Let G be a quasi-core-p p-group with G p + 1 = 1 and p > 2 . Then exp ( G ) p .
Proof. 
Suppose the result is not true and G is a counterexample of minimal order. For any g 1 , g 2 G , let H = g 1 , g 2 . By Lemma 1, ( g 1 g 2 ) p = g 1 p g 2 p x , where x 1 ( H ) H p . Since c ( H ) < c ( G ) , H p = 1 . By induction, exp ( H ) p and so exp ( 1 ( H ) ) = 1 . Thus x = 1 . It follows that there exist a , b G such that o ( [ a , b ] ) > p and exp ( G 3 ) p .
By induction, we may assume G = a , b , [ a , b ] = c and L = a , c . Then, according to Lemma 1, we see [ a p , b ] = c p y , where y 1 ( L ) L p . Since c ( L ) < c ( G ) , L p = 1 and exp ( L ) p . Thus y = 1 . Since G is a quasi-core-p p-group, a p G . So c p a . It follows from Theorem 1 that o ( c ) = p 2 . Similarly, we see c p b .
Without loss of generality, we may assume a b = a p s = b p t , a p s = b p t and s t 2 . If s > t , then, by letting b 1 = a p s t b , we see [ a , b 1 p ] = c p and c p b 1 p , in contradiction to the hypothesis. So s = t . Let b 2 = a b 1 . Then, by Lemma 1, we see b 2 p = a p b p z , where z 1 ( G ) G p . Since G = c , [ c , g ] | g G , we see 1 ( G ) = c p . Then 1 ( G ) G p Z ( G ) and exp ( 1 ( G ) G p ) p . Thus o ( z ) p and o ( b 2 ) = p s . Noticing that [ a , b 2 p ] = c p , we see c p b 2 p . If s = 2 , then c p = b 2 p , which implies b 2 p = a p b p z Z ( G ) , a contradiction. If s > 2 , then c p = b 2 p s 1 = a p s 1 b p s 1 . It follows that a b = a p s 1 , another contradiction. □
Corollary 4.
Let G be a quasi-core-p p-group and exp ( G p + 1 ) = p n with p > 2 and n 0 . Then exp ( G ) p n + 1 .
Proof. 
If n = 0 , then the conclusion holds by Lemma 9. Thus we may assume n 1 . Set G ¯ = G / G p + 1 . Then G ¯ p + 1 = G p + 1 ¯ = 1 ¯ . It follows from Lemma 9 that exp ( G ¯ ) p , which implies exp ( G ) p n + 1 . □
Corollary 5.
Let G be a quasi-core-p p-group and c ( G ) = p + n with p > 2 and n 0 . Then exp ( G ) p n + 1 .
Proof. 
If n = 0 , then the conclusion holds by Lemma 9. Thus we assume n 1 . Set G ¯ = G / G p + n . Then c ( G ¯ ) = p + n 1 . By induction, we see exp ( G ¯ ) p n . Since G p + n = [ G p + n 1 , G ] Z ( G ) , by Lemma 9, we see exp ( G p + n ) p . It follows that exp ( G ) p n + 1 . □
Theorem 5.
Let G be a quasi-core-p p-group with p > 2 . If G is abelian, then exp ( G ) p 2 and exp ( G 3 ) p .
Proof. 
Suppose that the result is not true and G is a counterexample of minimal order. Then there exist a , b G such that o ( [ a , b ] ) p 3 . We may assume G = a , b , [ a , b ] = c and L = a , c . By Lemma 1, [ a p , b ] = c p x , where x 1 ( L ) L p . By induction, exp ( L ) p 2 and so exp ( 1 ( L ) ) p . On the other hand, since [ a , c ] p Z ( G ) , it is easy to see that exp ( L 3 ) p . So o ( x ) p . According to Theorem 1, we see o ( c p x ) = p , which implies o ( c ) p 2 , in contradiction to the hypothesis. So exp ( G ) p 2 . Thus, for any g G , we see g p Z ( G ) . It follows that exp ( G 3 ) p . □
Theorem 6.
Let G be a quasi-core-3 3-group. Then exp ( G ) 9 and exp ( G 3 ) 3 .
Proof. 
Take a , b G with o ( a ) 9 and o ( b ) 9 . Let K = a , b . Then, by Lemma 1, ( a b ) 3 = a 3 b 3 c , where c 1 ( K ) K 3 . Since K G 4 , we see c ( K ) 3 by Theorem 3. Thus exp ( K ) 3 by Corollary 5, which implies o ( c ) 3 . It follows that ( a b ) 9 = a 9 b 9 = 1 . So, we may assume d ( G ) = 2 . According to Corollary 5 and Theorem 3, we see exp ( G ) 9 .
Take x G and y G . Then o ( x ) 9 and so x 3 Z ( G ) . Assume [ x , y ] = z and L = x , z . Then, by Lemma 1, 1 = [ x 3 , y ] = z 3 w , where w 1 ( L ) L 3 . Since L G 5 Z ( G ) , by Lemma 9, we see 1 ( L ) L 3 = 1 . It follows that z 3 = 1 . For any g , h G 3 with o ( g ) 3 and o ( h ) 3 , then, by Theorem 3, we see [ g , h ] G 6 = 1 . So o ( g h ) 3 , which implies exp ( G 3 ) 3 . □

4. Quasi-Core-2 2-Groups

In this section, we investigate the quasi-core-2 2-groups.
Lemma 10.
Let G = a , b be a non-abelian metacyclic quasi-core-2 2-group with a G and o ( a ) = 2 n . Then [ a , b ] = a 2 n 1 , a 2 or a 2 + 2 n 1 .
Proof. 
Since G is a non-abelian metacyclic 2-group, we see n 2 and G is one of the groups listed in Lemma 4.
If G is a group listed in (1) in Lemma 4, then the conclusion holds by the classification of p-groups with a cyclic maximal subgroup.
If G is a group listed in (2) in Lemma 4, then G = a , b | a 2 r + s + u = 1 , b 2 r + s + t = a 2 r + s , [ a , b ] = a 2 r with r 2 and u r . We may assume s + u 2 . By calculation, it is easy to see [ a , b 2 ] = a 2 r + 1 . Since G is a quasi-core-2 2-group, we see a 2 r + 1 b 2 , which implies s 1 . Let a 1 = a b 2 t . If s = 0 , then a 1 a = 1. It follows from G is quasi-core-2 that a 1 2 Z ( G ) , which implies a 2 Z ( G ) . However, it is impossible. If s = 1 , then o ( a 1 ) = 2 r + 1 and [ a 1 2 , b ] = a 2 r + 1 a 1 2 . It follows that a 2 r + u = a 1 2 r , which implies b 2 r + t a . It is also impossible. So s + u = 1 and therefore [ a , b ] = a 2 n 1 .
If G is of type (3) in Lemma 4, then G = a , b | a 2 r + s + v + t + u = 1 , b 2 r + s + t = a 2 r + s + v + t , [ a , b ] = a 2 + 2 r + v with r 2 and u 1 . It follows from [ a , b 2 ] b that s + t 1 and so s + t + u 2 . We may assume s + t + u = 2 and so u = s + t = 1 . Then b 2 r + s + t = a 2 n 1 and [ a , b ] = a 2 + 2 n 2 . We assume o ( b ) = 2 m . If r + s + t = 2 , then, since ( b a ) 2 = b 2 a 2 n 2 , we see o ( b a ) = 4 . On the other hand, [ a , ( b a ) 2 ] = a 2 n 1 . So, by the hypotheses, we see a 2 n 1 ( b a ) 2 = b 2 a 2 n 2 , a contradiction. If r + s + t 3 , then o ( b 2 m 3 a 2 n 3 ) = 4 and [ b , ( b 2 m 3 a 2 n 3 ) 2 ] = a 2 n 1 . Thus a 2 n 1 ( b 2 m 3 a 2 n 3 ) 2 = b 2 m 2 a 2 n 2 , another contradiction. So the conclusion holds. □
Corollary 6.
Let G be a quasi-core-2 2-group. Then Φ ( G ) is abelian and 2 ( G ) G Z ( G ) .
Proof. 
For any a , b G , we may assume H = a 2 , b is not abelian and o ( a ) = 2 n . By the hypotheses, we see a 2 G and so H is metacyclic. It follows from Lemma 10 that [ a 2 , b ] = a 2 n 1 , a 4 or a 4 + 2 n 1 . Then, it is easy to see that [ a 2 , b 2 ] = 1 , which implies Φ ( G ) is abelian.
Take g G with g 4 G . Then [ g 2 , h ] Ω 1 ( g ) for any h G , which implies [ g 4 , h ] = 1 and therefore g 4 Z ( G ) . So 2 ( G ) G Z ( G ) . □
Corollary 7.
Let G be a quasi-core-2 2-group. Then, for any a G , G / C G ( a 2 ) C 2 × C 2 , G / C G ( a 4 ) C 2 and if G / C G ( a 4 ) C 2 , then a 4 G and a Z ( G ) = Ω 1 ( a ) .
Proof. 
Without loss of generality, we may assume a 2 Z ( G ) , o ( a ) = 2 n and n 3 . By Corollary 6, we see Φ ( G ) C G ( a 2 ) , which implies G / C G ( a 2 ) is elementary abelian. For any g G / C G ( a 2 ) , according to Lemma 10, we see [ a 2 , g ] = a 4 , a 2 n 1 or a 4 + 2 n 1 . It is easy to see that G / C G ( a 2 ) C 2 × C 2 and G / C G ( a 4 ) C 2 . If G / C G ( a 4 ) C 2 , then, there exists an element b G C G ( a 4 ) such that [ a 2 , b ] = a 4 . So a 4 G and a Z ( G ) = Ω 1 ( a ) . □
Lemma 11.
Let G be a quasi-core-2 2-group with c ( G ) = 2 . Then exp ( G ) 4 .
Proof. 
If not, then there exist a , b G such that o ( [ a , b ] ) 8 . We may assume [ a , b ] = c . Then [ a 2 , b ] = c 2 . By induction, o ( c 2 ) 4 and so o ( c ) = 8 . It follows from Lemma 10 that c 2 = a 4 , which implies a 4 Z ( G ) . However, [ a 4 , b ] = c 4 1 , a contradiction. So the conclusion holds. □
Theorem 7.
Let G be a quasi-core-2 2-group with c ( G ) = n and n 2 . Then exp ( G ) 2 2 ( n 1 ) .
Proof. 
If n = 2 , then the conclusion holds by Lemma 11. Thus we may assume n 3 . Set G ¯ = G / G n . Then c ( G ¯ ) = n 1 . By induction, we see exp ( G ¯ ) 2 2 ( n 2 ) . Since G n = [ G n 1 , G ] Z ( G ) , by Lemma 11, we see exp ( G n ) 4 . It follows that exp ( G ) 2 2 ( n 1 ) . □
Theorem 8.
Let G be a non-abelian quasi-core-2 2-group with d ( G ) = 2 . Then 1 ( G ) , G 4 are cyclic, and either G Z ( G ) C 2 × C 2 × C 2 or G = a , b | a 8 = 1 , a 4 = b 4 = c 2 , [ a , b ] = c , [ c , a ] = [ c , b ] = 1 .
Proof. 
If G is metacyclic, then the conclusion holds by Lemma 10. So we may assume G = a , b is non-metacyclic, [ a , b ] = c , o ( a ) = 2 n , o ( b ) = 2 m and o ( c ) = 2 t with n m . Thus G = c , [ c , g ] | g G . By Corollary 6, Φ ( G ) is abelian. So [ c , g ] 2 = [ c 2 , g ] c 2 , which implies 1 ( G ) c 2 and therefore 1 ( G ) is cyclic. Now we consider the following two cases: c ( G ) = 2 and c ( G ) > 2 .
Case 1.
c ( G ) = 2 .
By Lemma 11, we see exp ( G ) 4 . We may assume exp ( G ) = 4 . Then o ( c ) = 4 and [ a 2 , b ] = [ a , b 2 ] = c 2 . Thus n m 3 and c 2 a b . Without loss of generality, we may assume a b = a 2 u = b 2 v , a 2 u = b 2 v and u v 2 . Let b 1 = a 2 u v b . Then [ a , b 1 2 ] = c 2 . If u > v or v 3 , then o ( b 1 ) = 2 v . Thus c 2 = b 1 2 v 1 , which implies a 2 u 1 b , a contradiction. So u = v = 2 and a 4 = b 4 . Noticing that G = a , b 1 and [ a , b 1 ] = c , we see a 4 = b 1 4 by the above. It follows from o ( b 1 ) = 8 that o ( a ) = 8 . So, we see G = a , b | a 8 = 1 , a 4 = b 4 = c 2 , [ a , b ] = c , [ c , a ] = [ c , b ] = 1 .
Case 2.
c ( G ) > 2 .
In this case, we consider the following two subcases: G is cyclic and G is not cyclic.
Subcase 1.
G is cyclic.
If o ( c ) 4 , then c 2 Z ( G ) and G Z ( G ) C 2 . So we may assume t 3 . By Lemma 10, we see [ c , a ] = 1 , c 2 , c 2 + 2 t 1 or c 2 t 1 . If [ c , a ] = c 2 , then exp ( G Z ( G ) ) = 2 . Thus we may assume [ c , a ] = c 2 t 1 and [ c , b ] = 1 . It follows that [ a 2 , b ] = c 2 + 2 t 1 . According to Lemma 10, it is easy to see c 2 = a 4 . So [ a 4 , b ] = 1 and therefore o ( c ) 4 , in contradiction to the hypothesis.
Subcase 2.
G is not cyclic.
Since [ a , b ] = c , [ a 2 , b ] = c 2 [ c , a ] . By Lemma 10, we see [ c , a ] = c 2 a 4 , c 2 a 4 + 2 n 1 , c 2 or c 2 a 2 n 1 . Similarly, [ c , b ] = c 2 b 4 , c 2 b 4 + 2 m 1 , c 2 or c 2 b 2 m 1 . It follows that G c , a 4 , b 4 , [ [ c , a ] , G ] 1 ( [ c , a ] ) and [ [ c , b ] , G ] 1 ( [ c , b ] ) . Then [ G 3 , G ] 1 ( G 3 ) 1 ( G ) . So G 4 is cyclic.
Now we prove exp ( G Z ( G ) ) = 2 . Assume [ c , a ] = c 2 a 4 or c 2 a 4 + 2 n 1 , and n 4 .
If [ c , b ] = c 2 , then G = c , a 4 . Since G is not cyclic, we see [ c , a ] 1 . Take g G Z ( G ) and assume g = c 2 i a 4 j . It follows from [ g , b ] = 1 that o ( g ) 2 . So exp ( G Z ( G ) ) = 2 .
If [ c , b ] = c 2 b 2 m 1 , then G = c , a 4 , b 2 m 1 . If [ c , a ] = 1 , then a 4 c and G = c , b 2 m 1 . It is easy to see that exp ( G Z ( G ) ) = 2 . Assume [ c , a ] 1 . Take h G Z ( G ) and assume h = c 2 k a 4 l . It follows from [ h , b ] = 1 that o ( h ) 2 and so exp ( G Z ( G ) ) = 2 .
If [ c , b ] = c 2 b 4 or c 2 b 4 + 2 m 1 , we may assume m 4 by the above. It is easy to see that a 8 , b 8 c . Thus [ b 8 , a ] = 1 , which implies o ( b ) = 16 and b 8 = a 2 n 1 . On the other hand, we see [ ( a 2 n 3 b 2 ) 2 , a ] = b 8 and therefore b 8 = a 2 n 2 b 4 . It follows that [ a , b 4 ] = 1 . However, it is impossible.
Assume [ c , a ] = c 2 or c 2 a 2 n 1 . Without loss of generality, we may assume [ c , b ] = c 2 or c 2 b 2 m 1 . Then G c , a 2 n 1 , b 2 m 1 . It is clear that exp ( G Z ( G ) ) = 2 . □

Author Contributions

Both authors have contributed to this paper. Writing-original draft, J.W. and X.G., Writing-review and editing, J.W.

Funding

This research was funded by the research project of Tianjin Sino-German University of Applied Sciences grant number [313/X18015] and [309/JG1742].

Acknowledgments

The authors would like to thank the referee for his or her valuable suggestions and useful comments which contributed to the final version of this paper.

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Buckley, J.T.; Lennox, J.C.; Neumann, B.H.; Smith, H.; Wiegold, J. Groups with all subgroups normal-by-finite. J. Aust. Math. Soc. 1995, 59, 384–398. [Google Scholar] [CrossRef]
  2. Lennox, J.C.; Smith, H.; Wiegold, J. Finite p-groups in which subgroups have large cores. In Proceedings of the Infinite Groups 1994, International Conference, Ravello, Italy, 23–27 May 1994; de Gruyter: Berlin, Germany, 1996; pp. 163–169. [Google Scholar]
  3. Cutolo, G.; Khukhro, E.I.; Lennox, J.C.; Wiegold, J.; Rinauro, S.; Smith, H. Finite quasi-core-p p-groups. J. Algebra 1997, 188, 701–719. [Google Scholar] [CrossRef]
  4. Cutolo, G.; Smith, H.; Wiegold, J. On core-2 2-groups. J. Algebra 2001, 237, 813–841. [Google Scholar] [CrossRef]
  5. Huppert, B. Endliche Gruppen I; Springer: Berlin, Germany, 1967. [Google Scholar]
  6. Berkovich, Y. Groups of Prime Power Order, Volume I; Walter de Gruyter: Berlin, Germany, 2008. [Google Scholar]
  7. Xu, M.Y.; An, L.J.; Zhang, Q.H. Finite p-groups all of whose non-abelian proper subgroups are generated by two elements. J. Algebra 2008, 319, 3603–3620. [Google Scholar] [CrossRef]
  8. Newman, M.F.; Xu, M.Y. A note on regular metabelian groups of prime-power order. Bull. Austral. Math. Soc. 1992, 46, 343–346. [Google Scholar] [CrossRef]
  9. Xu, M.Y.; Zhang, Q.H. A classification of metacyclic 2-groups. Algebra Colloq. 2006, 13, 25–34. [Google Scholar] [CrossRef]

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Back to TopTop