Questions tagged [p-groups]

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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Reference Request : Representation Theory Of Finite p-groups

In the (xi) group of the classification of groups of order $p^4$ given by W.Burnside in his book," Theory of Groups Of Finite Order". The group ($\mathbb{Z_{p^{2}}}\rtimes \mathbb{Z_{p^{}}}) ...
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28 views

Finitely Presented Group of order $p^4$

In the (xi) group of the classification of groups of order $p^4$ given by W.Burnside in his book," Theory of Groups Of Finite Order". The group ($\mathbb{Z_{p^{2}}}\rtimes \mathbb{Z_{p^{}}}) ...
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1answer
16 views

Meta cyclic p-group

While studying meta cyclic p groups, I came across an interesting class of meta cyclic groups which can be written as semi-direct product of two cyclic p-groups of order $p^m$ and $p^n$ respectively. ...
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1answer
46 views

On classification of groups of order $p^5$

Can someone suggest me some source where the author has classified all non-isomorphic groups of order $p^5$ ? Edit 1 : I need complete classification (not upto isoclinism), and also in finitely ...
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28 views

Structure of non abelian finite p-groups

I am familiar with the concepts of direct products, semi-direct products, wreath products and central products of groups. After seeing the classification of finite $p$-groups upto order $p^4$,(Theory ...
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56 views

Help to understand a step in the fix point theorem

Let $G$ be a $p$-group and $X$ be a finite set on which G operates. We define $X_G:=\{x \in X: g \circ x=x \; \forall g \in G\}$ set the of all fixed points. For the proof of $|X|\equiv |X_G|$ mod $p$ ...
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1answer
52 views

What is the maximum order of the elements in the group of order ${p^4}$ with 3-generator?

Let say we have a group of order ${p^4}$ with 3-generator $\langle x \rangle$, $\langle y \rangle$ and $\langle z \rangle$ where $|\langle x \rangle|={p^2}$, $|\langle y \rangle|=p$ and $|\langle z \...
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29 views

Extra special $p$-groups

I am looking for a reference for the following theorem: Let $G$ and $H$ be two extra special $p$-groups with the same order and the same exponent. Then $G\cong H$. Thank you in advance
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155 views

How to prove the number of the elements of order $p$ in a group of order $p^4$ is ${p^3}-1$

This is a follow up to this question. Let $G = \langle x,y,z\mid{x^p} = {y^p} = {z^{p^2}} = 1,[x,y] = z,[x,z] = [y,z] = 1\rangle$. To find the number of elements of order $p$, can i use the same ...
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112 views

How to find the number of elements of order $p$.

This is a follow up to this question. Let $G = \langle x,y,z\mid{x^{{p^2}}} = {y^p} = {z^p} = 1,{x^y} = {x^{1+p}},[x,z] = [y,z] = 1\rangle$. By euler phi function, the number of elements of order $...
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90 views

Let $G$ be a nonabelian group of order $p^{3},$ where $p$ is a prime. Show that $G$ has exactly $p^{2}+p-1$ distinct conjugacy classes.

Problem Let $G$ be a nonabelian group of order $p^{3},$ where $p$ is a prime. Show that $G$ has exactly $p^{2}+p-1$ distinct conjugacy classes. Attempt Let $G$ be a nonabelian group with $|G|=...
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60 views

Prove that for any positive integer $m\leq n$ there is a normal subgroup $H$ of $G$ such that $|H|=p^m$. [closed]

Let $G$ be a group with order $p^n$, where $p$ is a prime number and $n$ is a positive integer. Prove that for any positive integer $m\leq n$ there is a subgroup $H$ of $G$ such that $|H|=p^m$. Does ...
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66 views

Group of order $q^3p^3$, where $p,q$ are twin primes greater than $10$, is solvable

Let $q>p>10$ be twin primes, i.e., $q=p+2$. Show that every group of order $q^3p^3$ is solvable. This should be proven without using Burnside's theorem. Looking at the Sylow $p$-subgroup and ...
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A simple group with $|\operatorname{Syl}_p⁡ G| \le 6$ is cyclic

Let $G$ be a simple, finite group, s.t. for every prime $p$, it satisfies $k_p=\left|\operatorname{Syl}_p⁡ G\right| \le 6$. Show that $G$ is cyclic. My attempt: Let $n=p_1^{e_1}p_2^{e_2}\ldots p_r^{...
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39 views

Let $G$ be a p-group. Let $H$ be a proper subgroup of $G$. Show that there exists $g$ $\in$ $G \setminus H$ such that $gHg^{-1}=H$.

Let $G$ be a p-group. Let $H$ be a proper subgroup of $G$. Show that there exists $g$ $\in$ $G \setminus H$ such that $gHg^{-1}=H$. I tried to use a counting argument. Let's assume by contradiction ...
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65 views

Group of order 81 and exponent 3

I know that there is one group only of order 81 which is non abelian and of exponent 3. But I have no idea how to prove it. This question is personal. I looked at Wims, which suggested the answer. ...
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1answer
44 views

How many subgroups does a Prüfer group have?

Let $Z_{2^\infty}:=\{z\in \mathbb C:z^{2^n}=1,$for some $n\in \mathbb N\}$. This is a countable group. But I am not sure about the cardinality of the set of its subgroups. Does it have uncountably ...
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43 views

Let $p$ be a prime number and let $G$ be a finite $p\text{-group}$. Let $M$ be a maximal subgroup of $G$.

QUESTION: Let $p$ be a prime number and let $G$ be a finite $p\text{-group}$. Let $M$ be a maximal subgroup of $G$. Show that $M$ is a normal subgroup of $G$ and that $| G: M | = p$. THE HINT GIVEN ...
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52 views

If $|G|=p^n$, then for any $H<G$, there is a $K\leq G$ such that $H\lhd K$ and $[K:H]=p$

I have the following exercise: Problem. Group $G$ has order $p^n$, where $p$ is a prime number and $n$ is a positive integer. Prove that for any proper subgroup $H<G$ there is a subgroup $K\leq ...
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64 views

Semidirect product and group action

I want to understand the following lemma: Let $G$ be a finite group satisfying $G = P \rtimes F$, where $P$ is a cyclic $p$-group for some prime $p$, $|F| > 1$ and $(p, |F|) = 1$. Then each ...
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31 views

Proving $P$ is a Sylow $p$-group of $PN$

I am having trouble solving the following problem: Let $G$ be a finite group of order $p^an$, where $p$ is a prime and $p \nmid n$. Let $P$ be a Sylow $p$-group in $G$ and let $N \unlhd G$. It can ...
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42 views

Some results on derivation of a group

I am trying to get a complete proof of the following result (Ref. https://link.springer.com/article/10.1007/s00605-016-0938-5 Page-682, Lem.2.7): ${\mathrm{\bf Definition:}}$ Let $G$ be a group and $...
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1answer
32 views

A presentation of group of order $p^3$ and exponent $p$

I am trying to verify that the so-called Heisenberg group over the finite field ${\mathbb{F}}_p$ of order $p$ (with order $p^3$ and exponent $p$) has the following presentation: $$ G = \langle x, y ~:~...
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1answer
39 views

Cardinality of direct product of Sylow $p$-subgroups

Let $G$ be a finite group such that, for all prime number $p$, $P_p$ is a normal Sylow $p$-subgroup of $G$. Let $I$ denote the set of prime numbers dividing $|G|$ and $$K=\bigcup_{n\in\mathbb{N}}\{g\ |...
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1answer
28 views

Is the inverse image of a $p$-group under the canonical homomorphism also a $p$-group?

Let $G$ be a finite group and $H$ a normal subgroup of $G$. Let $f:G\rightarrow G/H$ be the canonical homomorphism. Let $Q\leq G/H$ be a $p$-subgroup of $G/H$. I have to show that $f^{-1}(Q)\leq G$ ...
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50 views

Quotient of quotient groups and Sylow $p$-subgroups

Let $G$ be a finite group and $N$ a normal subgroup of $G$. Let $\pi:G\rightarrow G/N$ be the canonical homomorphism. Suppose $P$ is a Sylow $p$-subgroup of $G$. Then $\pi[P]\leq G/N$. Note that $G$ ...
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42 views

Unique subgroup of index p in an inifite p-group

A question that recently came up in my homework: Let $p$ be a primer number and let $G$ be a $p$-group. Prove that if $G$ has a unique subgroup of index $p$ it must be that $G$ is cyclic. I know how ...
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28 views

Subgroup generated by Sylow $q$-groups of a finite group

Let $G$ be a finite group. Let $\mathfrak{P}$ denote the set of prime numbers and $n$ the order of $G$. Since $n>0$, there exists a unique family $(\nu_q(n))_{q\in\mathfrak{P}}$ of elements of $\...
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38 views

Every finite group contains a Sylow $p$-subgroup

Let $G$ be a finite group of order $n$ and $p$ a prime number. Write $n=p^rm$ for some $r\in\mathbb{N}$ and $m\in\mathbb{N}_{\geq1}$ such that $m\not\in p\mathbb{Z}$. Let $$E=\{X\subset G\ :\ |X|=p^r\}...
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46 views

A cyclic subgroup of a $p$-group

Let $G\ne\{e\}$ be a $p$-group. Then there exists $x\in Z(G)$ such that $x\ne e$. Let $k>0$ such that $x^{p^k}=e$: i.e. let $p^k$ be the order of the element $x$. I would like to show that the ...
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1answer
99 views

$G$ has a unique normal subgroup of order $p$ iff $G$ is cyclic center.

Let $G$ be a p-group. Proof that $Z(G)$ is cyclic if and only if $G$ has a unique normal subgroup $H$ of order $p$. I am supposed to prove it without using Sylow theorems. I already prove the first ...
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1answer
42 views

A condition implying powerfulness of finite $p$-groups, where $p$ is odd

For an odd prime $p$, a finite $p$-group is called powerful if $[G,G] \subseteq G^p$, where $G^p = \langle g^p ~:~ g \in G \rangle$. Prove that : $[G,G] \subseteq [G,G,G] G^p$ implies $G$ is powerful. ...
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85 views

Pattern in Number of Conjugacy Classes of p-groups

I was playing around with the number of conjugacy classes of $p$-groups in GAP and made the following conjecture: If there is a group of order $p^{2n}$ with $k$ conjugacy classes then there is a ...
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1answer
38 views

Classifcation of $p$-groups with cyclic subgroup of index $p$

I am trying to read the Classification of p-Groups with cyclic subgroup of index p done in Cohomology of Groups by Brown. He starts his proof by construction the exact sequence, $0→\mathbb{Z}_q→G→H→0$ ...
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1answer
57 views

A $p$-group that is divisible.

Suppose $G$ is an infinite $p$-group and $H$ is a minimal infinite subgroup of $G$ (where $p$ is a given prime). If $H=pH$ then $H$ is divisible. How can this be? An element of $H$ will only be ...
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49 views

Are there any references on extensions $G$ of a cyclic group $C_2$ by $2$-groups $P$?

Are there any references on extensions $G$ of a cyclic group $C_2$ by $2$-groups $P$ such that $1\neq a\in C_2$ is a square element in $G$? In other words, if $G/{C_2}\cong P$, where $P$ is a 2-group,...
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110 views

Finite intersection property for sets containing generating elements of derived subgroups of quotients

What I need to prove is a consequence of the following theorem. Theorem A. Let $G$ be a finite $p$-group and suppose that its derived subgroup $G'$ is generated by 2 elements. Then there exists $x\...
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1answer
59 views

Are there examples of subgroups of $\Bbb Z[\frac1n]/q\Bbb Z$ not totally ordered by inclusion?

I have the theorem that the Prufer P-groups (of which $\Bbb Z[\frac12]/\Bbb Z$ is one example) are the only infinite groups whose subgroups are ordered by inclusion. That this property holds for $\...
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1answer
51 views

On generalized fitting subgroup

I can't understand so much the second paragraph (page 160) of the proof of the lemma 31.17(1) (pages 160) in M. Aschbacher, Finite Group Theory about generalized fitting subgroup. Here I post the ...
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1answer
41 views

Order of conjugacy class of a p-group

I' m trying to solve the following exercise: Prove that, if G is finite of order $p^n$, p prime, $n\geq3$ and $|Z(G)|=p$, then G contains a conjugacy class of order $p$. I know that every class of ...
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3answers
100 views

If $P\le G$ is a sylow-$p$ and $Q$ is any $p$ subgroup, then $Q\cap P = Q\cap N(P).$

If $P\le G$ is a sylow-$p$ and $Q$ is any $p$ subgroup, then $Q\cap P = Q\cap N(P).$ I'd appreciate any help. I have a proof from some old notes but it says that it is sufficient to prove that if $...
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2answers
71 views

Equivalent definitions of a $p$-group

In Dummit and Foote, p.139, a $p$-group is defined as a group of order $p^\alpha$ for $\alpha \geq 1$. I also found online a definition of a $p$-group as a group in which every element has a power of $...
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1answer
65 views

Let $G$ be a group with $p^3$ elements, $p$ prime. Find the cardinal of the set $\{\operatorname{C}(x)\mid x\in G\}.$

Let $p$ be a prime number and $(G,\cdot)$ be a group with $p^3$ elements. We denote by $\operatorname{C}(x)$ the centraliser of $x\in G$. If $|\operatorname{Z}(G)|=p$, then find the cardinal of the ...
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2answers
78 views

Abelian group, invariant and p-group

For an abelian group $G$ I have shown that $$ G=Z(p^{l_1})\times...\times Z(p^{l_r})\Rightarrow G^p\cong Z(p^{l_1-1})\times...\times Z(p^{l_r-1}) \, \text{and ord} (G/G^p)=p^r. $$ A $p$-group $G$ is ...
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70 views

A question about the Frattini argument and normalizers

$G$ is a finite group and $N$ is a normal subgroup of $G$. Let $H/N$ be any nontrivial subgroup of $G/N$ of prime power order. Then we have $|H/N|=p^n$, for some prime $p$ and $n\geq 1$. Let $P$ ...
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5answers
85 views

About an order of a p-group [duplicate]

I'm trying to show that if G is a Group, then $|G| = p^2 \Rightarrow G$ is abelian. The path I'm taking relies on supposing that $|Z(G)| = p$ and forming the quotient group $\overline{G} = G/Z(G)$. ...
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2answers
119 views

Q: how to describe these results by a descendants tree in gap

I wrote an implement to find the "fullyInvariantGroups" in GAP and the results appeared as below: ...
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1answer
62 views

Finite groups with elements of order a prime power

Consider a finite group $G$ where the order of each element is a power of a certain prime number $p$, then $G$ is a $p$-group. My question: are there groups that are not $p$-groups, but for which the ...
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1answer
50 views

Multiplicative group of order $2^k$ has proper subgroup containing set of all squares

Specifically, I'm trying to solve the following problem: Let $G$ be a multiplicative group of order $2^k$ where $k\geq1$. Show that $G$ has a proper subgroup $H<G$ containing the subset $S=\{g^2:...
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1answer
43 views

normalizer tower question

Let $P$ be a finite $p$-group and $Q$ a proper subgroup of $P$. Define the normalizer tower of $Q$ in $P$ as follow: \begin{equation} N^0(Q) = Q \mbox{ and } N^i(Q) = N_P(N^{i-1}(Q)) \end{equation} ...

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