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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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Number of conjugacy classes of a finite p-group

Let $p$ be a prime number and $G$ be a non-Abelian finite $p$-group. Is it true that the number of conjugacy classes of $G$ is not a power of $p$?
2
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1answer
50 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 $p$-...
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0answers
23 views

Bounding the exponent in unit groups of modular group algebras

Let $p$ be a prime number, $G=:G_0$ a (non-Abelian) finite p-Group and $K$ a finite field with $\operatorname{char}(K)=p$. It is well-known that the group $G_1:=1+\operatorname{rad}(KG)$ is a p-group ...
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0answers
6 views

Isomorphism problem for the center of modular group algebras

Let $p$ be a prime number, $G,H$ a finite p-Group and $K$ a finite field with $\operatorname{char}(K)=p$. It is well-known that the group $1+\operatorname{rad}(KG)$ is a p-group containing $G$. My ...
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0answers
9 views

Defect of subnormality in modular group algebras

Let $p$ be a prime number, $G$ a finite p-Group and $K$ a finite field with $\operatorname{char}(K)=p$. It is well-known that the group $1+\operatorname{rad}(KG)$ is a p-group containing $G$. $G$ is ...
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0answers
6 views

A question to the ascending central chain in modular group algebras

Let $p$ be a prime number, $G$ a finite p-Group and $K$ a finite field with $char(K)=p$. It is well-known that the group $1+rad(KG)$ is a p-group containing $G$. Let us focus on the sequence $G\cdot ...
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1answer
25 views

Centralizers in p-groups

Let $G$ be a finite p-group and $g\in G$ such that $C_G(g)=C_G(g^p)$. Is it possible that for all $x\in G\setminus C_G(g)$ the identity $x^{\langle g^p \rangle}=x^{\langle g \rangle}$ is valid? E.g.,...
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3answers
552 views

A simple question on characteristic subgroups

Suppose $P$ is a finite abelian $2$-group and $U$,$V$ are characteristic subgroups of $P$ such that $|V:U|=2$. Does it follow that $P$ has a characteristic subgroup of order $2$?
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0answers
168 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. ...
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1answer
22 views

Group as direct sum of cyclic groups

What are necessary conditions for a cyclic group $G$ to be a direct sum of cyclic groups? I saw somewhere that $G$ must be a non $p$-group. But I couldn't prove it. Thank you for your hints/help
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42 views

No simple groups of given order.

I am trying to show the following: Prove that there are no simple groups of the given order: 42. 200. 231. 255. I understand that they need to be broken down into their prime factors. I was ...
2
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1answer
41 views

Either $H \triangleleft G$ or exist a conjugated subgroup $H^g \subseteq N_G (H)$, in which $g \in G$, with $H^g \neq H$.

Let $G$ be a $p-$group. If $H$ be a subgroup of $G$, prove that either $H \triangleleft G$ or exist a conjugated subgroup $H^g \subseteq N_G (H)$, in which $g \in G$, with $H^g \neq H$. In my opinion,...
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0answers
14 views

Showing that a group of prime order is in the center of a $p$-group. [duplicate]

Let $N $ be a normal subgroup of order $p$ contained in a group $G$ of order $p^n$. Here $p$ is a prime number. Then I have to show thst $N$ is in the center of $G$. This is an exercise 15 in p.92 of ...
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1answer
35 views

Abelian $p$-group and proof of the existence of cyclic subgroup

Consider the theorem Let $G$ be a finite Abelian group with order $|G|=p^n$ and $a$ an element of maximal order in $G$, then there is a subgroup $H$ of $G$ such that $G\cong |a|\times H$. I'm ...
2
votes
1answer
98 views

Non-abelian Groups of Order $p^3$

I am trying to show that a non-abelian group $G$ of order $p^3$ is isomorphic to one of two groups constructed on page 48 of these group theory notes (see examples 3.14 and 3.15 on that page). Here ...
3
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1answer
61 views

Let $G$ be a $p$-group of nilpotency class at most 2, where $p$ is an odd prime. Then ${\{x^p| x \in G\}}$ is a subgroup of $G$.

Let $G$ be a $p$-group of nilpotency class at most 2, where $p$ is an odd prime.Then $A=\{{x^p| x \in G}\}$ is a subgroup of $G$. As $G$ is a group of nilpotency class at most 2, if the nilpotency ...
4
votes
1answer
74 views

Proving this Hall algebra is commutative without Matlis duality

For a finite abelian $p$-group $G$ we have that $$ G \simeq \mathbf{Z}/(p)^{\lambda_1} \oplus \dotsb \oplus \mathbf{Z}/(p)^{\lambda_r} $$ for some positive integers $\lambda_1 \geq \dotsb \geq \...
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0answers
28 views

Let $G$ be a group of order $p^n$ where $p$ is a prime and $n \in \mathbb{N}$. Prove that exist normal subgroups [duplicate]

Let $G$ be a group of order $p^n$ where $p$ is a prime and $n \in \mathbb{N}$. Prove that exist normal subgroups $N_{1},N_{2},N_{3},...,N_{n}$ with $|N_{i}|=p^i$ for all $i \in ${$1,2,3,...,n$}.
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3answers
616 views

Prove, that group of order $p^2$ is abelian.

I know there is a proof using these theorems: The center of a finite p−group is non-trivial For any group G , $G/Z(G)$ is cyclic iff $G$ is abelian, or in otherwords: the quotient $G/Z(G)$ can never ...
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0answers
65 views

Does non-abelian $\Phi(G)$ imply, that $p^5 | |G|$ for some prime $p$?

Suppose $G$ is a finite group, such that $\Phi(G)$ is non-abelian. Does there always exist such prime $p$, that $p^5 | |G|$? Here $\Phi$ stands for Frattini subgroup. Using the same method, as the ...
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1answer
28 views

Rank of a $p$-group

Here I've found two definitions of the rank of a $p$-group https://groupprops.subwiki.org/wiki/Rank_of_a_p-group. However, for the $2$-group $\mathbb{Z}/\mathbb{Z}_{4}$, the rank with the first ...
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votes
1answer
32 views

$p$-group problem

Let $A,B,C$ are three subgroups in a way that $1<A \triangleleft B \triangleleft C$. With $B/A$ and $C/B$ are $p$-groups. Then prove that $|C|$ is also a $p$-group. I have been trying to prove it ...
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3answers
44 views

Cauchy's theorem proof clarification (group theory)

Cauchy's theorem says that if $G$ is a finite group with $p | |G|$ (when $p$ is prime), then $G$ contains an element of order $p$. When following the proof from wikipedia: https://en.wikipedia.org/...
4
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1answer
37 views

Maximal subgroup of Wreathed 2-groups

Definition A $2$-group $S$ is called wreathed if it is isomorphic to $(C\times C)\rtimes \langle i \rangle$ where $C$ is a cyclic group of order $2^n$ and $i$ is an involution with action $(a,b)^i=(...
5
votes
1answer
46 views

Let $G$ be group with order $p^n$; does there then exist a sequence of normal subgroups?

I would like to show the following statement: Let $p$ be a prime. Let $G$ be group with order $p^n$. Let $H$ be a normal in $G$ with order $p^k$. Then prove $H$ has subgroups $K$ such that $K$ has ...
2
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1answer
116 views

Computing the class-preserving automorphism group of finite $p$-groups

Let $G$ be a finite non-abelian $p$-group, where $p$ is a prime. An automorphism $\alpha$ of $G$ is called a class-preserving if for each $x\in G$, there exists an element $g_x\in G$ such that $\alpha(...
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1answer
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Order of elements of the Prüfer groups $\mathbb{Z}(p^{\infty})$ [duplicate]

Let $\mathbb{Z}(p^{\infty})$ be defined by $\mathbb{Z}(p^{\infty}) = \{ \overline{a/b} \in \mathbb{Q}/ \mathbb{Z} / a,b \in \mathbb{Z}, b=p^i$ $ with$ $ i \in \mathbb{N} \}$, I wish show that any ...
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3answers
3k views

In a group $G$ where $\exists!$ nontrivial, proper subgroup, show that $G$ is cyclic and $\lvert G\rvert=p^2$, for $p$ prime.

Suppose that $G$ is a group that has exactly one nontrivial proper subgroup. Then we have to show that $G$ is cyclic and order of $G$ is $p^2$ where $p$ is prime. I tried as, if $a$ and $b$ two ...
5
votes
1answer
97 views

If $G$ is a $p$-group then $\Phi(G)=G'G^p$

Okay this problem is quite the confounding one for me. If $G$ is a $p$-group then it follows that $\Phi(G)=G'G^p$. Where: $\Phi(G)$- Frattini subgroup (which in this case is the intersection ...
1
vote
1answer
49 views

If $\operatorname{class}(G) = 2$ and $\exp(G) = 4$ then $\exp(G') = 2$?

Let $G$ be a finite $p$-group. I'd like to prove (or disprove) that if the nilpotency class of $G$ equals two (i.e., $1 \neq G' \le Z$, where $Z$ is the center of $G$) and the exponent of $G$ equals ...
2
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1answer
139 views

Let $G$ be a non-abelian finite $p$-group, do we have $Z(G) \leq \Phi(G)$ in general?

Let $G$ be a non-abelian finite $p$-group. I wonder if $Z(G) \leq \Phi(G)$, where $\Phi(G)$ is the Frattini subgroup of $G$? I'd known that it is true for non-abelian groups of order $p^3$ but I ...
4
votes
2answers
91 views

Finding the number of subgroups of $\mathbb{Z}_{p^3} \oplus \mathbb{Z}_{p^2} $.

What is the number of subgroups of $\mathbb{Z}_{p^3} \oplus \mathbb{Z}_{p^2} $? I'm not really sure how to figure it out. I tried seeing subgroups of each $\Bbb Z_{p^n}$ but I'm not sure I'm going to ...
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0answers
44 views

If $P$ is a Sylow $p$-subgroup of $G$ then there exists a normal subgroup $K$ such that $G/K$ is isomorphic to $P/P \cap G'$

In 'Finite Groups' by Gorenstein, it is stated that if $P$ is a Sylow $p$-subgroup of $G$ then there exists a normal subgroup $K$ such that $G/K$ is isomorphic to $P/P \cap G'$′. The proof is the ...
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1answer
45 views

About quotients of Lower Exponent$-p$ central Series

Let $G$ be a finite $p-$group of number of generators $d$ and exponent$-p$ class $c$, that is $c$ is the smallest integer satisfying $P_c(G) =1$ in the series $$ G=P_0(G) \geq ...\geq P_{i-1}(G)\geq ...
3
votes
1answer
93 views

Finding some explicit formula for $(ab)^n$ in any $a,b$ in a finite $p$-group.

If $G$ is a finite $p-$group, let $a$ and $b$ any two elements from $G$. Is there any formula for $(ab)^n$ involving $a^nb^n$ for any natural number $n$? That is, some formula like $(ab)^n = a^nb^...
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0answers
61 views

Intersection of all maximal subgroups of a finite group

Let $G$ be a group. Define $$F=\left\lbrace x\in G\;:\; \text{if } \left\langle S,x\right\rangle=G\text{ then }\left\langle S\right\rangle=G\right\rbrace.$$ I have that $F$ is a normal subgroup of $G$....
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1answer
39 views

Normal subgroup $N$ of a p-group $G$ intersects $Z(G)$ nontrivially; What is wrong with the following trivial argument?

I'm trying to show that a normal subgroup $N$ of a p-group $G$ intersects $Z(G)$ nontrivially (please don't tell how to show it), but it seem it is quite a trivial question considering the following ...
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0answers
38 views

On $n$th class-preserving automorphism of finite $p$-group

Let $G$ be a finite non-abelian $p$-group, where $p$ is a prime. An automorphism $\alpha$ of $G$ is called an $n$th class-preserving if for each $x\in G$, there exists an element $g_x\in \gamma_n(G)$ ...
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0answers
20 views

Quick question about proving that either H is contained in K or K is contained in H.

Let G be a cyclic p-group with subgroups H and K. Prove that either H is contained in K or K is contained in H. I am looking at Alan Wang's answer, and I am a little confused. Why is it $H\leq K$ and ...
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0answers
26 views

Why is the $p-$multiplicator of a $p-$group is an elementary abelian $p -$group?

Let $G$ be a $p-$group that have the finite presentation $F/R$($F$ is a free group of rank $d$); The $p-$multiplicator of $G$ is defined by $$ G^* = R/[F,R]R^p $$ Why $G^*$ in an elementary abelian $p ...
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2answers
34 views

Error in proving subgroup is normal

Let $G$ be a $p$-group and let $H$ be a subgroup of $G$ of index $p$ ($p$ is prime). Prove that $H$ is normal. I have tried to prove it, but accidentally have proven the opposite by some error: Let ...
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votes
0answers
35 views

Detailed usage of abelianisation and row-echelonisation?

Let $G=⟨x,y\mid[[x,y],x]=x^2,(xyx)^4,x^4,y^4,(yx)^3y=x⟩$ with $p=2$. I hope to show me in details how to compute $G/P_1(G)$(A polycyclic presentation for $G/P_1(G)$), where $P_1(G)$ is the second ...
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votes
2answers
40 views

Example of non-abelian groups with these properties

I am looking for examples of non-abelian groups of arbitrarily large size with the following properties Have order $p^a$, where $a$ is a positive integer and $p$ is prime. Contain an abelian ...
3
votes
1answer
60 views

Subgroups of $G^n$, where $G$ is a $p$-group

Let $G$ be a finite $p$-group and let $n > 0$. Let $G^n$ be the direct product of $n$ copies of $G$. Are all subgroups of $G^n$ isomorphic to $H_1 \times \dotsm \times H_n$ for some subgroups $...
1
vote
0answers
34 views

Relationship between Frattini subgroup and the first Agemo subgroup of a prime-power order group

Let $G$ be a $p$-group for an odd prime $p$. The Frattini subgorup $\Phi(G)$ is defined as the intersection of all maximal subgroups or, equivalently, as the subgroup of non-generating elements, i.e. ...
2
votes
0answers
33 views

Bound on order of commutator subgroup of a $p$-group

I was reading an article where it is claimed that If $G$ is a finite $p$-group with $|G|=p^n$ and nilpotency class of $n-2$ where $n\ge 7$ then $p\le|Z(G)|\le p^2$ and $p^{n-3}\le |G'|\le p^{n-2}$....
1
vote
1answer
18 views

Any $p$-order subgroup is normal in a $pk$ group

I'm looking to prove that if $G$ is a group of order $pk$ where $p$ is prime and $p>k$, that any subgroup $K\leq G$ of order $p$ is normal in G. Does anybody have any hints or tips for proving ...
1
vote
2answers
141 views

Number of normal subgroups of order $p^s$ of a $p$-group

Let $G$ be a $p$-group. Show that the number of normal subgroups of $G$ that have order $p^s$ is $1$ $mod(p)$. I think I have to use Sylow theorems and the fact that every subgroup of order $p^s$ is ...
1
vote
1answer
28 views

Are these two semidirect products isomorphic?

Let $p$ be a prime. Then there is a nonabelian semidirect product of $\mathbb{Z}_{p^2}$ and $\mathbb{Z}_p$. There is also a nonabelian semidirect product of $\mathbb{Z}_p\oplus\mathbb{Z}_p$ and $\...
8
votes
1answer
274 views

Unipotent action of pro-$p$-group

Say $p$ and $\ell$ are distinct prime numbers. Let $G$ be a pro-$p$-group which acts continuously on a finite-dimensional $\mathbb{Q}_\ell$-vector space $V$. Assume that the action of $G$ on $V$ is ...