# Questions tagged [p-groups]

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### Number of subgroups of a $p$-group of index $p^k$

Let $p$ be a prime, let $n$ and $k$ be positive integers and let $G$ be a group of order $p^n$. Further, let $a_{p^k}$ denote the number of subgroups of $G$ of index $p^k$. If $a_{p^k}$ is greater ...
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### Groups with a unique lonely element

Does there exist a finite group $G$ of order greater than two containing a unique element $g$ such that $$g\notin\langle x\rangle \hbox{ for all x\in G\setminus\{g\} ?}$$ Or we have another ...
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### Status of a conjecture of Thompson

Let $S$ be a finite group. Denote by $\mathcal{B}_0(S)$ the set of the subgroups $H$ of $S$ satisfying $|H:H'| > |K:K'|$ for every proper subgroup $K$ of $H$ ($H'$ denotes the drived subgroup of ...
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### Units in group rings

Let $F$ be any field with $p$ elements and $G$ be any finite $p$-group, combining together they form a group ring $FG$. And $V(FG)$ denotes group of units of coefficient-sum equal to 1 in $FG$. We ...
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### Structure/description of a finitely presented group

I am unable to see the structure of the following finitely presented group. $$\langle a,b,c,d : [a,b]=c=a^p,\ [c,b]=c^p=d^p,\ b^{p^2}=c^{p^2}=1 \rangle$$ I have tried in GAP, but it is not showing any ...
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### On isoclinism classes of finite p-groups

With reference to James, Rodney, The groups of order (p^6) ((p) an odd prime)., Math. Comput. 34, 613-637 (1980). ZBL0428.20013., My question is can we get isoclinism class $\phi_2$ for a finite p-...
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### On classifying groups of order $p^5$

Can someone suggest me some source where the author has classified all non-isomorphic groups of order $p^5$ ? I need complete classification (not upto isoclinism), and also in finitely presented form ....
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### Finite $p$-groups of co-class $3$, class at least $4$ and some controlled generator growth

I am trying to prove the following comment (Ref. https://link.springer.com/article/10.1007/s00605-016-0938-5 Page-684, Rmk3.2): Let $G$ be a finite $p$-group of co-class $3$, class $\geq 4$. Then $G$ ...
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### Direct proof (or reference) that a given $p$-group is extra-special

Writing a paper on algebraic surfaces, I was led to consider the finite group $\mathsf{H}(A)$ whose presentation is the following. I start with an anti-symmetric matrix $A=(a_{ij})$ of order $2n$ ...
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### An outer automorphism of a 3-group of maximal class

Let $G$ be a finite 3-group of maximal class. The center $Z(G)$ contains two elements other than the identity. Does there exist an endomorphism of $G$ that maps one of them to the other? This is true ...
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### $p$-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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### 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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### 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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### 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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### 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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### The nilpotency class and the derived length of a $p$-group

Let $G$ be a finite $p$-group of nilpotency class $c$ and of derived length $d$. As is well known, we have $d\leq \lfloor\log_2 c\rfloor+1$, (https://groupprops.subwiki.org/wiki/...
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Let $p$ be a prime number. Is there a formula for the number of subgroups of $$\mathbb{Z}/p\mathbb{Z}\times \mathbb{Z}/p^2\mathbb{Z}$$ $$\mathbb{Z}/p\mathbb{Z}\times \mathbb{Z}/p\mathbb{Z}\times\... 0answers 191 views ### Cyclic subgroups of finite p-groups Let G be a finite non-Dedekind p-group with non-cyclic center, where p is an odd prime. By [\langle x\rangle]_G=\{g^{-1}\langle x\rangle g\ |\ g\in G\}, I mean the conjugacy class of the ... 0answers 27 views ### Bounding the exponent of finite p-groups with normalizer conditions on cyclic subgroups Suppose P is a non-cyclic finite p-group satisfying the following two conditions: All cyclic subgroups of order p in P are normal (this is equivalent to saying that \Omega(P) \subset Z(P)). ... 2answers 732 views ### Existence of a cyclic non-normal subgroup in a p-group Let G be a finite non-abelian p-group, where p is an odd prime, N be a normal subgroup of G of order p, where \frac{G}{N} is non-abelian. Does there exist an element g\in G such that ... 1answer 667 views ### Extra special p-groups Let P be an infinite extra special p-group for some prime p, namely, Z(P)=P'=\Phi(P) and P/Z(P) is infinite elementary abelian. Let C be a Prufer q-group for some prime q\neq p. ... 1answer 672 views ### On  p -groups with at least one element of order  p^{2}  Let G be a finite non-abelian p-group such that G contains at least an element of order p^2 and for every nontrivial normal subgroup N, G/N has not any elements of order p^2 and G/Z(G) is ... 1answer 235 views ### p-groups with maximal class subgroup Suppose G is a finite non-abelian p-group of nilpotent class c. Is there a subgroup H of nilpotent class c and size p^{c+1}? If this is not true, is it possible to add some additional ... 1answer 183 views ### p-groups with special property on its centralizers Thanks for any help or comment. Suppose G is a finite non-abelian p-group. Suppose G has a proper non-abelian subgroup M such that for every non-central element x\in M, C_G(x)\subseteq M. ... 3answers 3k views ### Is there a nice explanation for this curious fact about cyclic subgroups? Here's something that I noticed that quite surprised me. Let G be a finite abelian group. Consider the following expression.$$ \nu(G) = \sum_{\substack{H \leq G \\ H \text{ is cyclic}}} |H| $$It ... 2answers 215 views ### p-groups as finite union of disjoint normal abelian subgroups I was interested in knowing if groups with following property have been studied( like what can be said about structure of the group) : "G can be written as disjoint union of a given number of ... 0answers 92 views ### Random pro-p groups via iterated uniformly random central extensions Inspired by this question on math.se, I want to understand the following construction of a random pro-p group: We want to construct an inverse system$$\cdots \xrightarrow{\alpha_i} G_i \...
I'm looking for examples of $p$-groups $G$ with the following three properties: the center of $G$ is $\mathbb{Z}/p$, and $G^{\text{ab}} = (\mathbb{Z}/p)^n$ for some $n$, and for every $g \in G$ whose ...
### Is $[729,57]$ a Sylow $3$-subgroup of some well-known group?
Let $G$ be the group $[729,57]$, using GAP's notation. I have so far two descriptions of the group: a presentation an embedding (not surjective!) of the group into a Sylow $3$-subgroup of the unit ...