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

Article
On Finite Quasi-Core-p p-Groups
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 . 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 , 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  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  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 .
Lemma 1.
(, 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.
(, 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.
(, 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.
(, 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.
(, 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]