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.

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-1
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0answers
20 views

proofing $Z(G)=\langle [x,u]\rangle$ if $M=C_G(u)$ is maximal subgroup

Let $G$ be non-abelian finite p-group, $p$ is odd, with cyclic center and $u\in G$ be of order $p$ if $M=C_G(u)$ (centralizer of $u$) be a maximal subgroup and $Z(G)\le M$ for $x\in G\setminus M$ how ...
0
votes
1answer
36 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
38 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
34 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
22 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
22 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
14 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
21 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
48 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
19 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
41 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
30 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
34 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
votes
0answers
23 views

Problem in abelian $p$-group

Let $H=\langle a,b,c\rangle$ be an abelian finite $p$-group such that $a$ has maximal order in $H$ $b\langle a\rangle$ has maximal order in $\frac{H}{<a>}$ I want to prove that $\exists ...
1
vote
1answer
61 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
51 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
22 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
191 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
33 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 ...
2
votes
1answer
29 views

If $A\lhd B\lhd C$ with $[C:A]=p^n$, is there $D\lhd A$ with $D\subset C$ and $D/A$ a finite $p$-group?

Let $A$ be a (possibly infinite) group. Consider subgroups $C\lhd B\lhd A$, and assume that $A/B$ and $B/C$ are both finite $p$-groups. Is there necessarily a subgroup $D$ normal in $A$ and ...
1
vote
1answer
46 views

under following conditions $G$ has only one subgroup of order $p$.

Let $|G|=p^m$ for $m \ge 2$. If every subgroup of $G$ of order $p^2$ is cyclic, then $G$ has only one subgroup of order $p$.
1
vote
2answers
77 views

On a classification of all the characteristic subgroups of a finite abelian $p$-group.

For any finite abelian group $G$, any $n\mid\exp G$ and any $m\mid\frac{\exp G}{n}$, let $nG[m]:=\{g\in nG\mid mg=0\}$. I wonder if every characteristic subgroup of a finite abelian $p$-group $P$ is ...
2
votes
1answer
45 views

$p$-group acting on a finite set

Let $G$ be a $p$-group. Prove that if $G$ acts on a finite set $X$ and $p$ does not divide $|X|$, then $X$ contains some element that is fixed by every element in $G$. Any thoughts? I'm stumped ...
3
votes
1answer
79 views

Difference between definitions of $p$-subgroup and Sylow $p$-subgroup

I'm reading Abstract algebra by Dummit and Foote and the following definitions are made: $1$. A group of order $p^{\alpha}$ for some $\alpha\geq1$ is called a $p$-group. Subgroups of $G$ which are ...
5
votes
1answer
57 views

Frobenius kernel is regular normal elementary abelian p-subgroup?

I'm attempting Exercise 3.4.6 in Dixon & Mortimer's book on Permutation Groups: Let $G$ be a finite primitive permutation group with abelian point stabilisers. Show that $G$ has a regular ...
3
votes
0answers
74 views

If a certain group action fixes every element $x$ such that $x^4=1$, then the action is trivial

This is a question from chapter $4D$ of Isaacs' Finite Group Theory. Let $A$ act via automorphisms on $G$, where $G$ is a $2$-group and $A$ has odd order. Show that if $A$ fixes every element $x$ in ...
1
vote
1answer
38 views

Automorphism group of a non_abelian p_group

Let G be a non abelian p_group. When is set of all automorphisms group of G a p_group?
1
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0answers
35 views

Number of the subgroups of a $p$-group with order $p^k$ is congurent to $1$ modulo $p$

Let $G$ be a $p$ group of order $p^n$ and $k\leq n$. Theorem:Number of the subgroups of $G$ with order $p^k$ is congurent to $1$ modulo $p$. I have found a proof of this theorem in Rotman's book ...
2
votes
1answer
40 views

Subgroups of Direct product of p-groups

I have to solve this problem: If $G =\text{Drp}(G_p)$ where $G_p$ is a $p$-group, $\text{Drp}(G_p)$ denotes the direct product of the $p$-primary components of $G_p$, and if $H < G$, prove that $H ...
6
votes
0answers
78 views

Abelian groups whose automorphism group is a $p$ group

$\def\Aut{\operatorname{Aut}}$ Let $G$ be a finite abelian group such that $\Aut(G)$ is an $p$ group ,that is, $|\Aut(G)|=p^n$ . Then can we determine the cyclic decomposition of $G$ or at least the ...
2
votes
1answer
37 views

Are Sylow p-subgroups in conjugate?

Let $G$ be an infinite group and $p$ be a prime number. Let $\mathscr{C}$ be a chain of p-subgroups of $G$ ordered by inclusion. Then, for every element of the union of $\mathscr{C}$ has an order ...
6
votes
1answer
107 views

An example of a simple infinite $2$-group

Is there an example of a simple infinite $2$-group? Informations If a $2$-group is Artinian I know that it also locally finite, so the simple $2$-group cannot be Artinian. Take the subgroup ...
4
votes
3answers
176 views

Group acting on its subsets

Let $ G $ be a group with $ |G|=mp^\alpha $ where $ \alpha\geq1 $ and p is prime integer with $p \nmid m$. Then denote the set of subsets of G, having $p^\alpha$ size, with $X$. Then with the action ...
5
votes
0answers
49 views

Reference request: groups of order $p^4$.

I am looking for a textbook or a paper which include the classification of groups of order $p^4$ ($p$ is prime) using generators and relations. In particular I like to understand which group $G$ ...
4
votes
2answers
63 views

Deduce that if $G$ is a finite $p$-group, the number of subgroups of $G$ that are not normal is divisible by $p$

Given: Let $G$ be a group, and let $\mathcal{S}$ be the set of subgroups of $G$. For $g\in G$ and $H\in S$, let $g\cdot H=gHg^{-1}$ Question: Deduce that if $G$ is a finite $p$-group, for some prime ...
1
vote
1answer
50 views

What does “exponent 2 nilpotency class 2” mean?

According to the book The Symmetries of Things, p. 208, the number of groups of order 2048 "strictly exceeds 1,774,274,116,992,170, which is the exact number of groups of order 2048 that have ...
0
votes
1answer
23 views

$|\{ x\in X: g.x=x \space\space\space \forall g\in G \}| = |X|\space mod \space p$

Let $G$ be a p-group. $|G|=p^n$ for some n. Let X be a finite set so that $\,p\nmid |X|\,$, G acts upon X. Denote $A:= \{ x\in X: g.x=x \space\space\space \forall g\in G \}$ I am trying to show ...
1
vote
0answers
34 views

Show the intersection of a nonidentity normal subgroup and the center of P is not trivial

P is p-group and M is a nontrivial normal subgroup of P. Show the intersection of M and the center of P is nontrivial. By the class equation, I proved that Z(P)is not 1. Then, how do prove I the ...
1
vote
1answer
48 views

Minimal number of relations for finite $p$-groups

From the (sharpened) Golod/Shafarevich inequality we know that for finite $p-$groups, where $r$ is the minimal number of relations and $d$ is the minimal number of generators, that $r > ...
0
votes
0answers
35 views

Order of center of a p-group deduce abelian [duplicate]

Let $G$ be a group of order $p^n$, $p$ a prime. Suppose the center of G has order at least $p^{(n−1)}$. Prove that G is abelian. Attempt: use the class equation $|G|=|Z(G)|+ \sum_{i \in ...
1
vote
1answer
61 views

Derived subgroup of a finite non-Abelian p-group is proper?

How do I show that the derived subgroup of a finite p-group is always proper? In Abelian groups, it's trivial. In non-Abelian groups, my intuition is that there should be some way to relate $G/Z(G)$ ...
1
vote
0answers
20 views

Index of subgroups in infinite p-group

If $G$ is a finite $p$-group, it is trivial that every subgroup has index $p^r$ for some integer $r$. If $G$ is infinite, this is not true as the index can be infinite. If $G$ is an infinite ...
0
votes
2answers
50 views

Part of simple proof of nontrivial center in p-group

I'm trying to understand the proof of a Burnside theorem (as stated in Beachy's Abstract Algebra p. 328): Let $p$ be prime number. The center of any $p$-group is nontrivial. Now, In the proof they ...
9
votes
2answers
282 views

A question on $p$-groups, and order of its commutator subgroup.

$\textbf{QUESTION-}$ Let $P$ be a p-group with $|P:Z(P)|\leq p^n$. Show that $|P'| \leq p^{n(n-1)/2}$. If $P=Z(P)$ it is true. Now let $n > 1$, then If I see $P$ as a nilpotent group and ...
4
votes
1answer
86 views

On finite 2-groups that whose center is not cyclic

Let $G$ be a finite 2-group such that $\left|\dfrac{G}{Z(G)}\right|=4$, $Z(G)$ is not cyclic and $Z(G)$ has at least one element of order 4. Then prove that there exists an automorphism $\alpha$ of ...
1
vote
1answer
60 views

Commutators Calculus

I was trying to understand the above Corollary but I have a problem, namely why in the second to last line $A_0 \leq \zeta_p(G)$? Any ideas? Definitions By recurrence we define $[x,_0\, y]=x$; ...
5
votes
2answers
66 views

Show finite group is $p$-group given some structure of group

Let $G$ be a finite group. If there exists an $a\in G$ not equal to the identity such that for all $x\in G$,$\phi(x) = axa^{-1}=x^{p+1} $ is an automorphism of $G$ then $G$ is a $p$-group. This is ...
4
votes
1answer
89 views

Finite Group with $n$-automorphism map

If $G$ is a finite group and $\phi(x) = x^{p+1}$ is an automorphism of $G$ with $order(\phi) |p$ then $G$ is a $p$-group...? If the order of $\phi$ is $1$ then $\phi(x) = x = x^{p+1} = x^px ...
0
votes
0answers
50 views

Show that 2 representations are not equivalent and find all the irreducible representations of G.

Show that 2 representations are not equivalent and find all the irreducible representations of $G$. The group $G=T_{16}$ has order 16 and presentation given by $G=\langle a,b : a^8=b^2=1, ...