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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Extra Special p -group

How to prove that in an extra special group of order p^{1+2n},the order of any abelian subgroup is of at most p^{1+n}?
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3answers
46 views

Image of p-group

Let $G$ be a group and $|G|=p^n$ for some prime $p$. If $f:G\to H$ is a surjective homomorphism, how do I know $H=f(G)$ also has cardinality a power of $p$?
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1answer
26 views

Local group rings

Let $k$ be a field of characteristic $p$ and $G$ a finite group. How do you prove that if $kG$ is local then $G$ is a $p$-group? (I know how to prove the converse but not this implication).
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1answer
18 views

The order of the normalizer of a $p$-subgroup of $S_{p}$ [closed]

I found it In Exercise in abstract algebra by Dummit and Foote. Let $P$ be a Sylow $p$-group of $S_p$. What is the order of $N_{S_p}(P)$?
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3answers
60 views

Group of order $p^2$ [duplicate]

Let $G$ be a group of order $p^2$ where $p$ is a prime. I want to show that either $G$ is cyclic or is the product of two cyclic groups of order $p$. After some work, the problem reduces to showing ...
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44 views

Proof of the fixed point theorem in group action.

$\textbf{Fixed point theorem:}$ Let $G$ be a $p$-group and let $S$ be a finite set on which $G$ operates. If the order of $S$ is not divisible by $p$, there is a fixed point for the operation of $G$ ...
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1answer
23 views

Finding a finite $p$-group of nilpotency class $n$ for each $n>1$

There's a problem in Rotman's group theory book that goes For $n\ge1$, let $G_n$ be a finite $p$-group of class $n$. Define $H$ to be the group of sequences $(g_1,g_2,\dots)$ for $g_n\in G_n$ and $...
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64 views

When can a dihedral group $D_{n}$ of order $2 n$ be a $p$-group?

A $p$-group is a group where the order of every group element is a power of the prime $p$. The presentation of a dihedral group $D_n$ of order $2 n$ is as follows. $$D_n = \langle x, y \mid x^n = y^2 ...
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0answers
23 views

Prove a function to be an automorphism of a $p$-group $G$.

Let $M$ be a maximal subgroup of a $p$-group $G$. For fixed $g\in G\backslash M$ and $z\in Z(G)\cap M$ of order $p$, the map \begin{align*} \alpha : G&\longrightarrow G\\ mg^i &\...
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1answer
32 views

How to find a normal subgroup in the $p-$group of order $p^5$ other than center?

Let $G$ be a non-abelian $p$-group of the order $|G|=p^5$ such that Frattini Subgroup $\Phi(G)$, Commutator Subgroup $G'$ and Center $Z(G)$ of $G$ are equal and Rank($G$)=$3$, Exponent($G$)=$p$ (there ...
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1answer
174 views

Number of $p$-groups with order $p^k$ and $|Z|=p$

What is known about the number of groups of order $p^k$ with $|Z|=p$ , which I denote $N(p^k)$ ? For $k=1$ , it is clear, that we have $N(p^k)=1$ : We have one group of order $p$ and since it is ...
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1answer
95 views

Random Group of order $4096$ with a center of size $2$

How can I create a random group of order $4096$ with a center of size $2$ ? The algorithm should be able to create every possible group with the given properties in principle. I think the list of ...
1
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1answer
31 views

Associated Lie algebras of p-groups of maximal class

I was reading a paper the other day on Lie algebras of maximal class and they keep saying that some results are taken from p-groups theory. So my question is how do you get the associated Lie ...
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1answer
62 views

$G$ a non-abelian group of order $p^3$. Show that $Inn(G)$ is abelian.

Let $G$ be a non-abelian group of order $p^3$ for prime $p$. Show that $Inn(G)$ is abelian. The center of $G$, $Z(G)$, is of order $p$ (can be seen in this question). I also know that $G / Z(G)=...
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0answers
21 views

About Special and Extra-special $p$-groups

A $p$-group $G$ is said to be special $p$-group if $Z(G)=[G,G]=$ elementary abelian. A $p$-group $G$ is said to be extra-special if $Z(G)=[G,G]=$ elementary abelian of order $p$. The questions ...
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48 views

Homomorphisms from a $p$-group to $\mathbb{F}_p$

I'm doing a problem on group cohomology and have reduced it to the following: if $P$ is a $p$-group then $\textrm{Hom}(P,\mathbb{F}_p) \simeq P/\Phi(P)$ where $\Phi(P)$ is the Frattini subgroup of $P$....
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1answer
69 views

why is a polycyclic group that is residually finite p-group nilpotent?

I am trying to solve an exercise in D. Robinson's book A Course in the Theory of Groups, which asks me to show that if $G$ is polycyclic and residually finite p-group for infinitely many prime p, then ...
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1answer
48 views

Show that $G$ is nilpotent [closed]

How do I show that $G$ is nilpotent given that if $G$ is polycyclic and $G$ is residually finite p-group?
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1answer
50 views

On finite capable $p$-group of class two

Do there exists a finite capable $p$-group $G$ of class two with cyclic center and the center is not subgroup of Frattini subgroup of $G$? A group $G$ is capable if there exists a group $H$ such ...
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1answer
38 views

Finite abelian p-group with an element of maximal order

I want to know, following theorem comes from which book? Theorem . Suppose $G$ is a finite abelian $p$-group and $a \in G$ has maximum order, then there exists a subgroup $K⊆G$ such that: $ \langle ...
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1answer
24 views

A finite $p$-group has a supersolvable series.

I'm being asked to show: Show a finite $p$-group $G$ has a supersolvable series, i.e. a normal series $$G=G_0\ge G_1\ge\cdots\ge G_m=1$$ such that each factor group is cyclic and each $G_i$ is ...
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1answer
30 views

There is a group representation of $\Bbb Z/p\Bbb Z$ that is not decomposable over the field $\Bbb F_p$, where $p$ is prime

let p be prime. prove there is a group representation of $\Bbb Z/p\Bbb Z$ that is not decomposable over a field $\Bbb F_p$ for similar and simpler questions i showed homomorphism $\mu$ such that $$\...
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40 views

Index of a maximal subgroup among normal abelian subgroups

Let $P$ be a $p$-group and $A$ maximal among abelian normal subgroups of $P$. Show that: 1) $A=C_P(A)$. 2) $|P:A|\mid (|A|-1)!$. 1) If $A$ is an abelian normal subgrup of a certain group $...
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0answers
30 views

Characteristic subgroups of non-abelian $p$-group

It is a difficult problem to determine all the characteristic subgroups of a non-abelian $p$-group. But, then, I would seek for as many characteristic subgroups as we can, for small order $p$-groups. ...
4
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1answer
60 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 ...
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0answers
16 views

Presentation for Special $p$-groups

Is a uniform presentation for special $p$ groups of rank 2 known? Thanks in advance.
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0answers
79 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. ...
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0answers
23 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 $G\to\text{Aut}(G)$ ...
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1answer
46 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 instances?...
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2answers
34 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
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1answer
39 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 $&...
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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.
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1answer
79 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 ...
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2answers
89 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 ...
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1answer
49 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 ...
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2answers
58 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 ...
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2answers
36 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 ...
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1answer
86 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, y^...
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1answer
58 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 ...
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41 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 $A(p)=\...
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0answers
66 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 the ...
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57 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 $p$-...
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2answers
124 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 ...
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0answers
31 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 ...
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1answer
42 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: ...
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1answer
96 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 ...
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1answer
31 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 $\...
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1answer
35 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 ...
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1answer
34 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 $...
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62 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 ...