# 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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### Quotient of a minimal non abelian $p$-group by its center

I am trying to solve the following question: Let $P$ be a nonabelian finite $p$-group in which every proper subgroup is abelian. Show that $P/Z(P) \simeq C_p \times C_p .$ Since $P$ is non-abelian I ...
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### Center of a nonabelian p group can be found by taking the intersections of centers of two suitable maximal subgroups

Let $P$ be a nonabelian $p$ group. Prove that there exist maximal subgroups $M$ and $N$ of $P$ such that $$Z(P)=Z(M) \cap Z(N).$$ I found that $Z(P) \subseteq Z(M) \cap Z(N).$ I have no idea for the ...
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### Conjugacy classes of a non-abelian group of order $p^3$

Let $G$ be a nonabelian group of order $p^3$, where $p$ is a prime. It's well known that $|Z(G)|=p$. The noncentral elements have all centralizer of order $p^2$, because $Z(G)<C_G(x)<G$ for ...
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### Prove/disprove: Let $G$ s.t $|G| = p^3 \implies |G'| \leq p$. [duplicate]

Let $G$ s.t $|G| = p^3$ for some prime $p$. Prove or disprove: $|G'| \leq p$. I couldn't think of any counter-examples so I started proving this and I'm stuck unfortunately and was hoping to seek ...
1 vote
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### 2-groups with abelianization of type (2,2)

On Olga Taussky's article "A Remark on the class field tower" there is a note on the last page where she states that P. Hall pointed out to her that exists 3 different groups of order $2^k$ ...
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### Cauchy's theorem for groups use unclear [duplicate]

Im reading about the Burnside's $p^aq^b$ theorem proof, and for the case when $b=0$ it uses that, since the order of the group is a power of a prime, by an elementary result of group theory we have ...
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### Proving G in a p-group using the centralizer of its elements

I came across this question in a past exam in an Algebraic structures course. It's a first course in group theory, and I had some difficulty solving this problem. Let $p \in \mathbb{N}$ be a prime ...
1 vote
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### Normalizer of subgroup of a finite $p$-group. [duplicate]

I have been struggling with the following exercise for a while now: Let $p$ be a prime number and $G$ be a finite $p$-group. Show for any real subgroup $H$ of $G$ that $H$ is a real subgroup of the ...
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### In a nonabelian group of order $p^4$, a maximal normal and abelian subgroup of $G$ is of order $p^3$.

Let $p$ be a prime and $G$ be a nonabelian group of order $p^4$. Let $H$ be a subgroup of $G$ maximal with respect to $H$ being normal in $G$ and $H$ being abelian. We have to show that $|H|=p^3$. I ...
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### Explicit construction of an outer isomorphism in a $p$-group.

Recently, I stumbled upon a theorem by Wolfgang Gaschütz (see below) that every non-trivial $p$-group which is non-trivial has an outer automorphism. However, the proof uses cohomology theory and as ...
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### If $G/G’$ is cyclic, prove by induction that $G/Z(G)$ is cyclic if $G$ is $p$ group. [duplicate]

Let $G$ be a $p$ group. If $G/G’$ is cyclic, prove $G/Z(G)$ is cyclic by induction. I tried playing with the base cases : if $|G|=p$ or $p^2$, $G$ is abelian, so $G/Z(G)=G/G =\{G\}$ which is ...
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### The Second Cohomology group with Coefficient in cyclic group of order 3.

I am computing the second cohomology group $H^2(G,\mathbb{Z_3})$ of $G=$SmallGroup(81,9). Its presentation is given by $G=\langle x,y,z\mid x^3=y^9=z^3=[y,z]=1,[y,x]=z,[z,x]=zy^3z^{-1}\rangle$. I ...
1 vote
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### Let $|G|=p^a$. Prove that if $|N|=p^{a-1}\Rightarrow N\lhd G$

Let $|G|=p^a$. Prove that if $|N|=p^{a-1}\Rightarrow N\lhd G$ If $a=1$ this is straightforward because $G$ is abelian and every group is normal, in particular $N$. Let us take $a>1$. First try: If ...
1 vote
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### Non-$p$-torsion difference of $p$-torsion elements in a $p$-group

I have a relatively simple question. For $p$ odd, does there exist a non-abelian $p$-group such that for every pair of non-commuting $p$-torsion elements $g$,$h$ their difference $g^{-1}h$ is not $p$-...
1 vote
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### Prove a simple module must be "trivial"

Let $P$ be a finite $p$-group, i.e. $|P|=p^n$ for some prime $p$. Let $F$ be a finite field of $p$ elements. Show that every simple $FP$-module trivial, where $FP$ is a group ring (which is actually a ...
1 vote
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### Image of augmentation ideal of a group ring under ring homomorphism.

Let $G,H$ be two finite $p$-groups and $f:\mathbb{F}_p[G]\to \mathbb{F}_p[H]$ a ring homomorphism of mod $p$ group rings. Is it always true that $f(I_G)\subset I_H$, where $I_G,I_H$ are the respective ...
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### Prove that $K’’=1$ for a particular $p$-group $K$

Let $S$ be a $p$-group, $N$ a normal subgroup of $S$ contained in $Z(S)\cap S’$. Consider $K$, another normal subgroup of $S$ with these two properties: $K/K’$ is elementary abelian $[K,K,K]=N$ How ...
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### How to show that $|G|=d_1^2+\cdots +d_s^2$ and $G$ is abelian?

In Representation Theory of Finite Groups: An Introductory Approach by Benjamin Steinberg, there is a corollary in which I feel confused. Corollary 6.2.6. Let p be a prime and let $G$ be a group of ...
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### What are all the finite groups where every nontrivial element has order $3?$

I know that the powers of any group where all elements have order $3$ also has this property. I also know that the Heisenberg group has this property and that all finitely generated such groups are ...
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### Group of order $2p$ where $p$ is an odd prime is isomorphic to either integers mod $2p$ or dihedral group of order $2p$.

$\newcommand{\Z}{\mathbb{Z}}$ Proposition: Let $p$ be an odd prime and $G$ a group of order $2p$. Then either $G \cong \Z/2p\Z$ or $G \cong D_{2p}$. Attempt: Well $G$ can be either cyclic or non ...
1 vote
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### Any group of order $p^2$ is abelian, where $p$ is prime; $G/Z(G)$ cyclic implies $G$ is cyclic. [duplicate]

I've taken an approach and it really feels like it should be getting me there, but I can't seem to close it off. Any hints? Attempt: If $G$ is a group of order $p^2$ then as a $p$-group $Z(G)$ is non-...
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### Showing that $|X| \equiv |\text{Fix}(G)|\pmod{p}$.

$\newcommand{\fix}{\text{Fix}}$ $\newcommand{\stab}{\text{stab}}$ $\newcommand{\orb}{\text{Orb}}$ Definition: Let $G$ act on a set $X$. We define $\fix(G) =\{x \in X : g \ast x = x, \forall g \in G\}$...
1 vote
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### Question on self-normalising $p$-subgroups of $p$-groups [duplicate]

Given a finite $p$-group $P$, we call a proper subgroup $Q<P$ centric if $C_P(Q)=Z(Q)$. The question is: can a centric subgroup $Q$ be self-normalising, which means that $N_P(Q)=Q$? I do not see ...
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### Smallest normal subgroup such that the factor group is a $p$-group

I am working with an Andreas Dress's article, and he says for a group $U$ (finite), the subgroup $U^p$ is the (well defined !) smallest normal subgroup of $U$ with $U/U^p$ a $p$-group. I think that ...
1 vote
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### Let $G$ be a group order order $p^3$. Show for any $g,h \in G$, we have $g^p h = hg^p$

Let $p$ be a prime number. Let $G$ be a group order order $p^3$. Show for any $g,h \in G$, we have $g^p h = hg^p$ If $|g|=1$, then we're done. If $|g|=p$, then we're done. If $|g|=p^3$, then group is ...
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### Prove that every group of order $p^m$ with $p$ a prime number can be generated by $m$ elements

Prove that every group of order $p^m$ can be generated by $m$ elements, where $p$ is a prime number. I am thinking about induction. Base case: if $m=1$, then $|G|=p$. So $G$ is cyclic. Hence, it's ...
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### Classification of nilpotent groups via $p$-groups

First some background: Some time ago I learned that since any finite abelian group is a direct product of (finite) cyclic groups, the finite cyclic groups ($\mathbb{Z}_n$) are the key to understanding ...
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### Let $G$ be finite & nonsimple such that all normal subgroups of $G$ are $p$-groups for a given prime $p$. Is it true that $G$ is also a $p$-group?

Let $G$ be a finite non-simple group such that all normal subgroups of $G$ are $p$-groups for a given prime $p$. Suppose also that the $p$-Sylow subgroups of $G$ are not normal. Is it true that $G$ is ...
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### How we illustrate the Prüfer-$3$ group elements on the unit circle?

I am trying to illustrate the Prüfer-$3$ group elements on the unit circle and I am not sure, if my approach is correct. It should be noted that I am mainly interested in the odd integers. Let us ...