# 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.

342 questions
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### 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 ...
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### 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 ...
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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### Proof of theorem: If $G$ is a finite nonabelian $p$-group, then $\mathrm{Aut}_c(G)=\mathrm{Inn}(G)$ if and only if $G′=Z(G)$ and $Z(G)$ is cyclic.

Consider the following theorem: If $G$ is a finite nonabelian $p$-group, then $\operatorname{Aut}_c(G)=\operatorname{Inn}(G)$ if and only if $G′=Z(G)$ and $Z(G)$ is cyclic. Notation $p$ is a ...
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### Show that the dihedral group $D_{16}$ is the internal direct product of its Sylow subgroups.

Show that the dihedral group $D_{16}$ is the internal direct product of its Sylow subgroups. (We use the notation $D_{16}$ for the dihedral group of order 32) Here's what I think. Since $D_{16}$ is ...
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### Central quotient of $p$-groups

Suppose $P$ is a finite $p$-group with center $Z(P)$ of order $p$. What kind of groups can appear as the central quotient $P/Z(P)$? For example, the quotient is in particular a so-called capable ...
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### Determine possible $p$-groups from center and quotient

Consider the following situation: I have given a finite $p$-group $P$ (in the case I am interested in $p = 2$) with cyclic center $Z(P)$ and I also know the structure of the quotient $P/Z(P)$ (which ...
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### Subgroup of $p$-group and cyclic center [closed]

This is the problem: Let $p$ be a prime and $G$ a $p$-group. Prove that $Z(G)$ (center of $G$) is cyclic if and only if $G$ has a unique normal subgroup of order $p$. I can't see why having a unique ...
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### Finite $2$-group with derived subgroup of order 8 [duplicate]

Does there exist a finite non-abelian $2$-group $G$ such that $G^{\prime}\cong D_8$ or $G^{\prime}\cong Q_8$? By an easy inspection with GAP, I could not find any example! Any answer or comment will ...