Use with the (group-theory) tag. Refers to questions concerning finite groups of prime power order or infinite p-groups such as Prüfer groups, pro-p-groups, and Tarski monsters. This tag is not for p-adic number systems.

learn more… | top users | synonyms

4
votes
1answer
54 views

Automorphisms of a group and subgroup

Is there a finite $p$-group $G$ such that $G$ has less number of automorphisms than some subgroup: $$|\mbox{Aut}(G)|<|\mbox{Aut}(H)| \mbox{ for some }H\leq G.$$ If there is such a group, then can ...
0
votes
0answers
13 views

Presentation for Special $p$-groups

Is a uniform presentation for special $p$ groups of rank 2 known? Thanks in advance.
3
votes
0answers
68 views

Classify all groups of order $p^2q^2$ up to isomorphism

Let $p,q \in \mathbb{N}$ be prime numbers with the properties $2 < p < q$ and $q - 1 , q + 1 \notin \left\langle p \right\rangle$ Classify all groups of the order $p^2q^2$ up to isomorphism. ...
1
vote
0answers
22 views

Homomorphisms $G\to\text{Aut}(G)$ for a $p$-group

I am interested in studying automorphisms of a $p$-group, or at least the highest power of $p$ dividing the order of the automorphism group, and feel like studying homomorphism $G\to\text{Aut}(G)$ ...
2
votes
1answer
44 views

Highest power of $p$ dividing $|\text{Aut}(G)|$ if $|G| = p^\alpha$

What is known about the highest power of a prime dividing the number of automorphisms of a $p$-group? I know that it is at least $p$ provided $p^2$ divides $|G|$, but can we say more in some ...
1
vote
2answers
26 views

What does “the subgroups of $G$ form a chain” mean?

I am being asked to show that: $G$ is a cyclic $p$-group $\iff$ its subgroups form a chain. What does "its subgroups form a chain" mean? Please keep in mind that I am just asking for the meaning of ...
3
votes
1answer
36 views

Number of $p$-groups of small order and of exponent $p$

In a very recent paper of MR Vaughan-Lee, it is proved that the number of $p$-groups of order $p^8$ and exponent $p$ is a polynomial (of fourth degree) in $p$. Let us consider $p$-groups of order ...
2
votes
0answers
40 views

On nonabelian $2$-groups of exponent $4$ and center $C_2$.

I am new to this forum, and would like to know whether it is possible to fully classify all nonabelian $2$-groups whose exponent is $4$ and center is $C_2$? Thanks.
7
votes
1answer
71 views

Power automorphisms which is not inner

I want to know an example of a small order non-abelian $p$-group $G$ with a power automorphism which is not inner, i.e. an automorphism of the form $g\mapsto g^k$ for all $g\in G$, but non-inner. In ...
4
votes
2answers
79 views

Classifying $2$-groups with cyclic center of order $2^n$ and index $4$

If $G$ is a $2$-group with cyclic center having order of $Z(G)=2^n$ and index of $|G/Z(G)|=4$, then how many groups of a particular order are possible? I know that there are two possibilities for ...
1
vote
1answer
32 views

When Center of group is a subset of Normalizer($Z(G) \subset N(a)$).

DEFINITION: If $a \in G$, then $N(a)$, the normalizer of $a$ in $G$, is the set $N (a) = \{ x \in G | xa = ax \} $. $Z(G)$ is the center of the group. I found the following proof - Lemma: If ...
2
votes
2answers
55 views

Known result about such $p$ groups

Let $G$ be a finite $p$ groups with, $|G'|=p$ $|G:Z(G)|=p^2$ $Z(G)$ is cyclic. $1)$ Can $G$ have nonabelian maximal subgroup ? It is clear that all maximals containing the center are abelian. Is ...
1
vote
2answers
28 views

A problem in decomposing a p group into direct sum of nontrivial subgroups

Hello all I have taken a group theory course where we are now covering p groups and we I have met the following exercise: Let $ G = Z/(p^n) $ is a(n Abelian) group of order $ p^n $ for a prime $ p ...
7
votes
1answer
75 views

Groups of order $64$ with abelian group of automorphism

G. A. Miller in 1913 constructed the first example of a non-abelian group of order $64$ with abelian group of automorphisms. It is the group $$G=(C_8\rtimes C_4)\rtimes C_2=\langle x,y,z\colon x^8, ...
1
vote
1answer
31 views

Frattini subgroup of a finite elementary abelian $p$-group is trivial

I would like to improve my proof of the following result: If $H$ is a finite, elementary abelian $p$-group, then $\Phi(H) = 1$. Here, $\Phi(H)$ is the Frattini subgroup, defined as the ...
0
votes
0answers
37 views

Given the primary decomposition for a finite abelian group, possible to deduce the p-group decomposition for said group?

Given the classification theorems as they can be found on https://en.wikipedia.org/wiki/Finitely_generated_abelian_group Given $A$ finite abelian group $|A|= \prod_{i=1}^r p_i^{n_i}$ and ...
2
votes
0answers
54 views

Unitriangular group $UT_n(\Bbb Z)$ is nilpotent with class $n$

The unitriangular group $UT_n(\Bbb Z)$ is the group of all $n \times n$ invertible triangular matrices with the identity on each entry of the main diagonal, and integer entries everywhere else in ...
0
votes
0answers
42 views

A trivial center for a p-group.

Can a p-group, where $p$ is a prime, have a trivial center? I'm supposed to consider the restricted wreath product of $C_p \wr T_p$, where $C_p$ is the cyclic group of order $p$ and $T_p$ is the ...
2
votes
2answers
95 views

Commutator Subgroup in a $p$-group

Let $G$ be a finite non-trivial $p$-group. Show that $G'$ (the commutator subgroup of $G$), is a proper subgroup of $G$. How could one show this result? I was thinking of first arguing that the ...
0
votes
0answers
28 views

On centers of infinite $p$-groups and nilpotent groups

If $G$ is a finite $p$-group, then its center is non-trivial, which forces that $G$ must be nilpotent. Consider infinite $p$-groups, i.e. infinite groups in which order of every element is some power ...
2
votes
1answer
29 views

On commutator relations and nilpotency class of a $p$-group

Suppose $G$ is a $p$-group, and $x,y\in G$ be arbitrary. If the commutator $[x,y]$ commutes with both $x$ and $y$, then the subgroup $\langle x,y\rangle$ has nilpotency class $\leq 2$. Question: ...
1
vote
1answer
73 views

Let $G$ be a p-group with $|G|=p^n$. Show $G$ has normal subgroups of order $p^m$ for each $0<m<n$.

Here is what I have so far. Statement: Let $G$ be a p-group with $|G|=p^n$. Then $G$ has normal subgroups of order $p^m$ for each $0<m<n$. The case $|G|=p^1$ is trivial. Proceed by induction ...
0
votes
1answer
27 views

problem with showing some relations in finite non abelian p-group

In some paper we have I don't get last two line, I don't know how to prove any of them. if you can give me some hint it would be good $\Omega_1=$ subgroup generated by all elements of order p ...
0
votes
1answer
28 views

about conjecture “every finite nonabelian p-group admits a noninner automorphism of order p”

by "The Kourovka Notebook. Unsolved Problems in Group Theory" there is a strong conjecture asserts that every finite nonabelian p-group admits a noninner automorphism of order p. What about finite ...
0
votes
1answer
31 views

Proving some prooperty in finite p-groups when $\theta_g$ is inner automorphism induced by element $g$

Let $G$ be a finite p-group $(p\ge3)$ and $\theta_g$ be the inner automorphism induced by element $g$. If $[G,g]\le Z_2(G)$ (upper central series) and $Z_2(G)$ is an abelian non-cyclic group of order ...
1
vote
0answers
55 views

example of p-group with infinite Frattini subgroup

Is there an infinite $p$-group $G$ with infinite Frattini subgroup $\Phi(G)\not = G$? In "Subgroups of Teichmuller Modular Groups" there is an example but I don't get it because I don't know much ...
1
vote
1answer
52 views

specific solution of Prove that $N_p(H) \ne H$

this is not a duplicate .. I'm trying to understand a specific answer of this question. the parts that I don't understand are marked in (1) , (2), (3) and explanations for what i don't understand are ...
0
votes
1answer
27 views

CA-group [abelian centralizer group]

I am searching for all information about CA-groups [abelian centralizer group] and i just found a German book [ Huppert ] and Nilpotent Centralizer group of Suzuki in 44 pages. I need more English ...
0
votes
0answers
40 views

find the factor groups of the p group - G and prove that G is solvable , where $|G|= p^a$ , p is prime

G is a p-group - $|G| = p^a$ p is prime i need to find the factor groups of G and prove that it is solvable. what i tried - EDITED: after watching the comments and investigating I know every ...
1
vote
0answers
63 views

on relation between upper central series and lower central series

Let $G$ be finite p-group of nilpotency class n and $d(G)=2$ with cyclic center $Z(G)=\langle z\rangle$ and $Z_2(G)$ is non-cyclic abelian group of order $p^3$ I want to understand relation between ...
0
votes
0answers
18 views

$\bar G=\frac{G}{\gamma_2G^p}$ is of class at most 2 when G is not powerful

Let $G$ be finite p-group of class c and order $p^n$ that is not powerful then $\gamma_2\not\le G^p$ (second lower central series is not contained in $G^p$) Let $\bar G=\frac{G}{\gamma_2G^p}$ How we ...
1
vote
1answer
32 views

property about centralizer of maximal subgroup

How we can show that for group $G$ (finite non-abelian p-group, I don't know which ones are necessary) and $M$ maximal subgroup of $G$. We can have $C_G(M)\le C_G(\Phi(G))\le Z(\Phi(G))$ $\Phi(G)$ ...
0
votes
1answer
47 views

Are subgroups of order $p^{n-1}$ maximal?

Let $G$ be finite p-group of order $p^n$, I know all maximal subgroups of order $p^{n-1}$ Is it right to say all subgroup of order $p^{n-1}$ are maximal subgroups? If not, what property $G$ must have ...
0
votes
1answer
41 views

If G is finite p-group then $d(G)=d(\frac{G}{\Omega_1(Z(G))})$

Let $G$ be a finite p-group such that $G$ has no non-inner automorphism of order p leaving Φ(G) elementwise fixed If $\Omega_1(Z(G))\le G'\le \Phi(G)$ how we can get $d(G)=d(\frac{G}{\Omega_1(Z(G))})$ ...
2
votes
1answer
64 views

looking for example of infinite p-group of nilpotency class 2

Is there infinite p-group of nilpotency class 2? If p=2 or p=3 would be better. and I prefer simple examples
2
votes
1answer
23 views

invariant properties between p-group and it's automorphism

Let $G$ be a p-group and $Aut(G)$ be group of automorphisms of $G$ which properties of $G$ can help us with studding $Aut(G)$? for example If $G$ is infinite/finite does this guaranty $Aut(G)$ be ...
1
vote
0answers
24 views

problem in understanding some part of proof of theorem

In proof of theorem: for every finite non abelian p-group $G$ of class 2 there is utter automorphism of calss 2 fixing $\Phi(G)$ by counterexample,If we know 1. $G'=\langle[a,b]\rangle$ 2. $H=\langle ...
-1
votes
1answer
18 views

The number of cyclic subgroups

Let $|G|=p^{n} $, then for every $d$, that $d|p^{n} $, there are cyclic subgroups of order $d$ for group $G$. These subgroups be as $G_{0} \subseteq G_{1} \subseteq ...\subseteq G_{n} =G$,where ...
1
vote
1answer
28 views

central series of $\frac{G}{Z_2(G)}$

Let $G$ be finite p-group I am trying to make central series for $\frac{G}{Z_2(G)}$ and more inportant what is nilpotency class of $\frac{G}{Z_2(G)}$ since ...
1
vote
0answers
53 views

Is $Z_2(G)$ abelian?

Let $G$ be non-abelian finite p-group and Let $\frac{Z_2(G)}{Z(G)}$ be elementary abelian group how we can get $Z_2(G)$ is abelian? If it's needed we also know $|Z(G)|=p$ and $|Z_2(G)|=p^3$
0
votes
0answers
20 views

Size of generating set of finite $p$-groups

For a group $X$, let $d(X)$ denote the minimum cardinality of a generating set for $X$. Let $G$ be a non-abelian finite p-group and $H\subset G$ a subgroup with $|H|=p$. Can we say that $d(H)=1$? If G ...
0
votes
0answers
17 views

$g^p$ in Center of G

Let $G$ is a $p$-group and $g^p\in Z(G)$ what we can say about $g$? in what kind of $G$ we can say $g\in Z(G)$?
1
vote
1answer
50 views

p-element centralizing a Sylow p-subgroup

Let $G$ be a finite group, $P$ a Sylow $p$-subgroup for a prime $p$ and $g$ a $p$-element with $gxg^{-1} = x$ for all $x \in P$. Then $g \in Z(P)$. Is this true? How can i prove that $g \in P$ ...
2
votes
0answers
47 views

p-group of class 2 with cyclic commutator subgroup

I was reading a paper "ON THE EXISTENCE OF NONINNER AUTOMORPHISMS OF ORDER TWO IN FINITE 2-GROUPS" you can get it from here. In Introduction page 2 He said "It is worth mentioning here that we need ...
0
votes
0answers
43 views

Let $ M $ be a maximal subgroup of a solvable group $ G $, and assume that $ G=MC $, for some cyclic subgroup $ C $.

Let $ M $ be a maximal subgroup of a solvable group $ G $, and assume that $ G=MC $, for some cyclic subgroup $ C $. Then $ \vert G : M \vert $ is a prime or $ 4 $. Also if $ \vert G : M \vert = 4 $ , ...
1
vote
1answer
66 views

need a 3-group with 3 generators

I need a $3$-group of nilpotency class $3$ with $3$ generators, but I can't find any group meeting these criteria. Does such a group exist? I've already used GAP-System as far as my PC can handle, ...
0
votes
0answers
56 views

A Rédei $p$-group is the union of its maximal subgroups

I read somewhere that if $G$ is a Rédei $p$-group, then $G$ is the union of its maximal subgroups. I want to know how I can prove this. By definition a Rédei $p$-group is a minimal finite non-abelian ...
2
votes
1answer
31 views

Center of a maximal subgroup

Is it possible to find a non-abelian group (preferably a finite $p$-group) with the following property? If M is a maximal subgroup of a group (or finite $p$-group) $G$, then $Z(M)\not \le Z(G)$.
1
vote
3answers
243 views

$G$ non abelian, order $p^3$ ($p$ prime). Suppose that the center is $p^2$, prove that $\exists\ x$ outside of the center, of order p

Let $G$ be a non abelian group of order $p^3$, with $p$ prime. I'm proving that $Z(G)$ (its center) is of order $p$. I already know how to do it by saying that its order can't be $p^3$, nor 1, and if ...
0
votes
1answer
48 views

Abelianization of a $p$ group.

Let $G$ be a $d$ generated finite $p$ group. Let $N$ be normal subgroup of order $p$ contained in $[G,G]\cap Z(G)$. Can we say say that $(G/N)/[G/N,G/N]=(G/N)_{ab}$ is a direct product of $d$ cyclic ...