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# Questions tagged [p-groups]

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### Relations of minimal number of generators

What is the command in GAP to find the all relations of minimal generators of a finite $p$-group $G$?
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### Do the class vector and character vector of a $p$-group determine each other?

To a finite $p$-group, we can associate two vectors $(v_0,v_1,\dotsc)$: The class vector - $v_i$ is the number of conjugacy classes of order $p^i$. The character vector - $v_i$ is the number of ...
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### How can I get my hands on McKay's “Finite p-Groups” lecture notes?

The notes I'm talking about are these. I emailed Peter Cameron, but he has since moved to a different university, and has no copies himself. I also emailed the school manager at Queen Mary, but they ...
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### For a pro-p, profinite group, abelianization being finitely generated is the same as being topologically finitely generated

I remember reading (without proof) that for $\Gamma$ a profinite, pro-$p$ group, the following are equivalent: 1) Every open subgroup $\Gamma_0$ is topologically finitely generated. 2) The ...
258 views

### Is the norm element characteristic in modular group rings?

Let $G$ be a finite $p$- group and let $\varphi$ be an automorphism of $\mathbb{F}_pG$ as $\mathbb{F}_p$-algebras and let $n = \sum_{g\in G} g$ be the norm element. Does it follow that $\varphi(n)=n$? ...
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### Defect of subnormality in unit groups of 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 $G$. $G$ is normal in $1+rad(KG)$ if and ...
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### about a strange property of p-groups of maximal class

I am trying to look for a finite $p$-group of maximal class of order at least $p^{2p+1}$ exponent at least $p^3$ which possibly has the following property : If s is an element in $G-G_1$ ($G_1$ is ...
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Let $G$ be a finite $p$-group of derived length $d$, which is not a Dedekind group. (i.e., possesses at least one non-normal subgroup). Let $G^{(d-1)}$ be the unique normal subgroup of $G$ of order $... 0answers 64 views ### p-group of maximal class I am trying to prove that if$G$is a$p$-group of maximal class and order$p^4$($p$odd), then its unique two-step centraliser$G_1=C_G([G,G])$is of the form$C_{p^2}\times C_p$. It is clear from ... 1answer 118 views ### Central extensions of Suzuki 2-groups Recall the definition of the finite Suzuki 2-groups: These are finite non-abelian 2-groups that contain more than one involution such that a solvable group of automorphisms permutes the involutions ... 1answer 163 views ### a question about finite 2-group Please help me about the following question: Suppose$G$is a finite 2-group and$x\in Z(M)\setminus Z(G)$for some maximal subgroup$M$of$G$such that$x^2\in Z(G)$, is it true$x\in Z_2(G)$? ... 0answers 118 views ### Conjugacy classes of non-normal subgroups of a finite$p$-group Let$G$be a finite$p$-group of derived length$d$and nilpotency class$c$. Suppose that$G$is not a Dedekind group (i.e., possesses at least one non-normal subgroup). Suppose that$G^{(d-1)}\$ has ...

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