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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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      134 views

      Normal subgroups of $p$-groups

      I was reading Professor Yukov Berkovich' paper "On Subgroups of Finite $p$-groups" when I stumled upon the following theorem: Let $G$ be a nonabelian $p$-group with cyclic Frattini subgroup, $|\Phi(...
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      1answer
      103 views

      $p$-groups with isomorphic subgroup lattices

      Given two non-abelian finite p-groups $P_1$ and $P_2$ of the same order that are not isomorphic. Can $P_1$ and $P_2$ have isomorphic subgroup lattices? (I'm not experienced with group theory, ...
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      125 views

      Computing the class-preserving automorphism group of finite $p$-groups

      Let $G$ be a finite non-abelian $p$-group, where $p$ is a prime. An automorphism $\alpha$ of $G$ is called a class-preserving if for each $x\in G$, there exists an element $g_x\in G$ such that $\alpha(...
      4
      votes
      1answer
      127 views

      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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      125 views

      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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      votes
      1answer
      88 views

      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 ...
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      254 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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      26 views

      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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      141 views

      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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      vote
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      104 views

      Certain $p$-group with cyclic center

      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 $...
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      0answers
      62 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 ...
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      votes
      1answer
      116 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 ...
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      1answer
      162 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 $x\in Z_2(G)$? ...
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      116 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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