# 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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### Question about a possible application of Burnside's basis theorem

Let $G=\langle a_1,\dots , a_d\rangle$ be a $d$-generator $p$-group of order $p^n$ (i.e. $d$ is the minimal number of generators). Further let $N$ be a characteristic and elementary abelian subgroup ...
• 291
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### How to check whether a finite $p$-group is regular in GAP?

I am trying to check whether a given $p$-group is a regular $p$-group in GAP. I am trying to use the command 'IsRegularPGroup(G)' for it. However I am getting 'Error, Variable: 'IsRegularPGroup' must ...
• 354
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### Questions on a proof on $p$-constrained groups

Theorem: Let $G$ be a group and $p \in \pi(G)$. Furthermore, suppose that $$\label{eq_p-constrained} C_{G/O_{p'}(G)}(O_p(G/{O_{p'}(G)})) \leq O_p(G/{O_{p'}(G)}).$$ If $P$ ...
• 167
1 vote
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### Proof of Thompsons $A \times B$-lemma

(Auxiliary lemma) Let $G$ be a $\pi$-group and $a$ a $\pi'$-element acting on $G$. If $X$ is a subnormal subgroup of $G$ with $[a, X] = 1 = [a, C_G(X)]$, then $[a, G] = 1$. Hey guys, I am having a ...
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### The order of center of any non-abelian $p$-group of order $p^n$

Let $G$ be a non-abelian $p$-group of order $p^n$ with $n>3$. In this paper there is (just after Lemma 2.10) a statement which says $p^2 \leq |Z(G)|\leq p^{n-2}$. I know that the center of a $p$-...
• 769
1 vote
104 views

### Counting homomorphisms from $S_n$ to a $p$-group

There are plenty of exercises and questions counting homomorphisms between groups. However, the following has not been asked yet and I can not see any way to count the homomorphisms using the common ...
• 2,802
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### The number of direct sum of elementary abelian 2-groups

Let $G=(Z_2)^n$, I want to know the number of direct sum of $G$($G=H \oplus K$) or a fine upper bound. For $G=Z_2 \oplus Z_2$, I have calculated that all of its direct sum decomposition is as follows: ...
• 347
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### MAGMA: How to efficiently do coercion of element of HomGrp to element of GrpAuto? [closed]

Suppose I have a finite $p$-group $G$ as GrpPC in MAGMA. The computation of the automorphism group $\mathrm{Aut}(G)$ takes a very long time. Suppose that I also ...
• 291
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### Huppert, III.19.2: How to construct a homomorphism from a $p$-group into the center of a maximal subgroup?

Let $G$ be a finite $p$-group and $N$ a maximal subgroup (so $G/N$ has order $p$) such that $Z(N) \leq Z(G)$. III.19.2 in Huppert's Book "Endliche Gruppen I" says that there exists a non-...
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### $p$-group with a cyclic subgroup

Throughout studying a paper about finite $p$-groups, I have the following question Let $G$ be a finite $p$-group with nilpotency class 3 and $\gamma_i(G)$ denote the i'th term of the lower central ...
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### 2-generated p-groups with generators of order p

Let $G=\langle a,b\rangle$ be a finite $p$-group such that $a^p=b^p=1$. Is there any result about the size of the set of $p$-elements $\Omega(G):=\{g\in G\mid g^p=1\}$? In particular, I'm interested ...
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### $p$-groups with nontrivial intersection of nonnormal subgroups

I study the following paper about finite $p$-groups, Finite groups in which the non-normal subgroups have nontrivial intersection, N. Blackburn, Journal of algebra, 3, 30-37 (1966). In this paper ...
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