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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.

This tag is for questions specifically pretaining to finite p-groups and their properties, such as isoclinism, schur covers and projective representations, and cohomology. The tag can also be used for questions about infinite p-groups such as Prüfer groups, pro-$p$-groups, or Tarski monsters. This tag is not for use with p-adic number systems, though it may be used for the $p$-adic integers $\displaystyle\varprojlim \mathbb{Z}/(p^n\mathbb{Z})$.

646 questions
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Two p-groups of exponent $p$ with same number of conjugacy classes but non-isomorphic centers

I am searching for an example of two $p$-groups of exponent $p$ with same number of conjugacy classes but not isomorphic centers. Background: The corresponding unit groups of the centers of the group algebras over $\operatorname{GF}(p)$ are…
Sven Wirsing
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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 $H =\text{Drp}(H\cap G_p)$. I have tried to do it in…
elisa
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characters in p-groups

I would like to know that some examples of p-groups with $|cd(G)| = dl(G)=3$, such that $cd(G)$ to denote the set of degrees of the irreducible characters of $G$ and $dl(G)$ to denote the derived length of $G$.
user115721
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Prove the order of $h$ when $h \in G\ \langle g \rangle$, $G$ is a finite abelian $p$-group that is not cyclic

The lemma to be proved is: Let $G$ be a finite abelian $p$-group that is not cyclic. Suppose that $g \in G$ has maximal order. If $h \in G \setminus \langle g \rangle$ has smallest possible order, then $|h| = p$. And the proof is written as: Let $g…
GalaxyY
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On the subgroup $\Omega_1(P)$ of a $p$-group

Let $P \cong \Bbb{Z}_{p^n} \times \Bbb{Z}_p$ be an abelian $p$-group of order $p^{n+1}$. We konw $\Omega_1(P)= \langle x \in P \mid x^p=1 \rangle$. Clearly, $\Omega_1(P) \cong \Bbb{Z}_{p} \times \Bbb{Z}_p$ and $P/ \Omega_1(P)$ is cyclic. Why all…
Rima
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