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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2answers
56 views

What is the center of a non-abelian group of order $3^5$? [on hold]

Suppose I have a non-abelian group $G$ of order $3^5$. I know then that $|Z(G)|$ must be one of $3,9,27,$ or $81$. Which of these are actually possible?
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1answer
28 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
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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
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1answer
43 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 $p$.
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2answers
68 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
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1answer
39 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
70 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
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1answer
55 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
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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 ...
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1answer
37 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?
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0answers
33 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
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1answer
38 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
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0answers
76 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
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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
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1answer
100 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
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3answers
171 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
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0answers
46 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
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2answers
60 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 ...
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1answer
47 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 ...
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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 ...
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0answers
30 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 ...
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1answer
46 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 > ...
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0answers
34 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 ...
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1answer
53 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) to ...
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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 ...
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2answers
46 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 ...
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2answers
257 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
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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
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1answer
59 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
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2answers
64 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
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1answer
87 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 ...
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0answers
48 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, ...
1
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1answer
81 views

Finite $p$-group in which all its maximal subgroups are cyclic

Let $G$ be a finite $p$-group, $|G|=p^n$. Let $M_1,\dots,M_r$ be all the maximal subgroups and suppose they are cyclic. Why is $\Phi(G)\le Z(G)$? $\Phi(G)$ is the Frattini subgroup. I have no idea ...
2
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1answer
69 views

On $p$-groups with a unique minimal subgroup

If $G$ is a finite group with a unique minimal subgroup, we know that $|G|=p^n$. I have to prove that if $p\neq2$ then $G$ is cyclic. This is the contest. What I don't understand is the following ...
3
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2answers
168 views

Cauchy's Theorem $\#$ of elements order $p$

In the stronger statement of Cauchy's Theorem it states the the number of elements of order $p$ is a multiple of $p$. http://en.wikipedia.org/wiki/Cauchy%27s_theorem_(group_theory)#Proof_2 I noticed ...
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1answer
41 views

Proving group is $p$-group by contradiction

http://www.proofwiki.org/wiki/Group_is_P-Group_iff_All_Elements_have_Order_Power_of_P Is $k$ a prime or a prime power? Sorry for this stupid question but I can't tell what $k$ is in this context ...
3
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2answers
42 views

$|G|=p^3$ non abelian $\Longrightarrow\gamma_2(G)\lneq Z(G)$

Let $G$ be a non abelian group of order $p^3$. Hence $c=2$ ($c$ is the nilpotence class of $G$). I'll write down some notation, in order to be clear. Let $Z_0(G):=1$, $Z_1(G)=Z(G)$, $Z_{k+1}(G)$ ...
6
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1answer
59 views

Classify $p$-groups in which all groups of the same order are isomorphic

The answer to “Are two subgroups of a finite $p$-group $G$, of the same order, isomorphic?” is definitely no. Such groups are very rare. How rare? Can you classify all finite $p$-groups $G$ such that ...
1
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1answer
50 views

$\exists a\in G-H$ such that $aHa^{-1}=H$

Let $G$ be a $p$-group with proper subgroup $H$. Show that there exists an element $a\in G -H$ such that $a^{-1} Ha = H$ Can you check my proof? Since $G$ and $H$ are $p$-groups their centers ...
3
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1answer
157 views

Subgroups of abelian $p$-groups

Let $A$ be an Abelian group of prime power order. It can be expressed as a (unique) direct product of cyclic groups of prime power order: $A = \mathbb{Z}_{p^{n_1}} \times \cdots \times ...
4
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1answer
124 views

Order and element set of the group with presentation $\langle a,b \;|\; a^9=1, b^3=a^3, [a,b]=a^3\rangle$

If we have a group presentation $G=\langle a,b \;|\; a^9=1, b^3=a^3, [a,b]=a^3\rangle$, how we will get the following values: The order of the group. The elements of the group written in terms of ...
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1answer
75 views

Sylow subgroups of soluble groups

Suppose $G \leqslant S_p$ acts transitively on $\{1,...,p\}$ for prime $p$. Let $P \leqslant G$ be a Sylow p-subgroup. Is it true that $G$ is soluble <=> $P \triangleleft G$?
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1answer
135 views

Lower Exponent P Central Series

The lower exponent $p$-central series for a $p$-group $G$ is defined by $G=P_1(G) > P_2(G) > \ldots > P_c(G) = 1$, where $$P_i(G)=[P_{i-1}(G), G] P_{i-1}(G)^p.$$ If $G_i=G/P_i(G)$ and ...
2
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0answers
217 views

number of elements of each order in p-groups $Z_{p^n}\rtimes Z_p$ and $Z_{p^n}\times Z_p$ [closed]

Do $p$-groups $\mathbb{Z}_{p^{n}}\rtimes \mathbb{Z}_p$ and $\mathbb{Z}_{p^{n}}\times \mathbb{Z}_p$ have the same number of elements of each order? (The prime $p$ is odd.)
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1answer
89 views

Classification of automorphism groups of groups of order $p^4$

For the purpose of classifying another algebraic structure which is parametrised by the choice of a group and of an automorphism I need the classification up to isomorphism of automorphism groups of ...
1
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1answer
40 views

about z(G) by concept of symplectic spaces

if G be a finite p-group and G' be isomorphic to zp, what we can say about z(G) by concept of symplectic spaces? is [G:z(G)] a perfect square? ((i take the elementary abelian group G/Z(G) as a ...
1
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1answer
38 views

Writing $p$-groups using $p$-adics

Is it possible to write any finite abelian $p$-group as $\mathbb{Z}_p^n/\mbox{im }(A)$ for some $n\times n$ matrix $A$ over $\mathbb{Z}_p$? Here $\mathbb{Z}_p$ denotes the $p$-adic integers.
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2answers
95 views

No group of following property. Is this true?

Let $p$ be a prime greater than 3 and $G$ be group of order $p^5$. Is it true that there is no group $G$ of order $p^5$ such that the order of frattini subgroup is $p^3$ and the order of center is ...
2
votes
2answers
118 views

Let $G$ be a finite $p$-group with has more than one maximal subgroup. Prove that $G$ has at least $p+1$ maximal subgroups.

Let $G$ be a finite $p$-group with has more than one maximal subgroup. Prove that $G$ has at least $p+1$ maximal subgroups. I don't have idea. Help me. Thanks in advanced. EDIT: I found a result ...
1
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1answer
117 views

Nonabelian group of order $p^4$ [closed]

Let $P$ be a nonabelian group of order $p^4$, where $p$ is a prime, and let $A$ be a subgroup of $P$ maximal with the property of being normal and abelian. Prove that $A$ is of order $p^3$. Thanks a ...