# Tagged Questions

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### p-group with large center

Is there any characterization for $p$-groups of order greater than $p^3$ which center has index $p^2$? (One group whit this property if $M(p^n)$)
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### Classification of automorphism groups of groups of order $p^4$

For the purpose of classifying another algebraic structure which is parametrised by the choice of a group and of an automorphism I need the classification up to isomorphism of automorphism groups of p-...
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### 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 $x\in M$, $C_G(x)\subseteq M$. ...
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### p-groups and 2-generated abelian images

Let $p$ be a prime number. Is there a finite nonabelian $p$-group $G$ such that any finite epimorphic $2$-generated image of $G$ is abelian?
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### A question on $p$-central $p$-groups

Let $p$ be a fixed prime. A group $G$ is termed $p$-central if every element of order $p$ in $G$ lies in the center. Having a finite $p$-group $G$ of rank $k$ (the least integer, such that every ...
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### A finite $p$-group with certain properties

Is there a finite $p$-group $G$ such that : (a) $G= \langle A,x,y \rangle$, with $G/Z(G)$ has exponent $p$, $A$ is a maximal abelian normal subgroup of $G$, and $G/A$ has order $p^2$ (thus it is ...
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### Number of Normal subgroups In a p-Group

Dear all, Does someone know of any paper/method that enables us counting/estimating the number of normal subgroups of some p-group of order $p ^n$ ($n$ is some natural number ? ) . Is there anyway ...
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### When can the restriction map be zero or onto?

Let $G$ be a finite $p$ group of order $p^n$ and $N$ be a subgroup of $G$ of index $p$. Let $res:H^2(G,A) \rightarrow H^2(N,A)$ be the restriction map on cohomology, where $A$ is a trivial $G$ module. ...
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### Is the Tensor/Exterior square $G\otimes G$ or $G\wedge G$ of infinite p-group also a p-group?

Let $G$ be an infinite countable p-group. Is it true that $G\otimes G$ or $G\wedge G$ are also p-groups? (where G acts on itself by conjugation). For simplicity, you can consider that $G=[G,G]$, and ...
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### Involutive automorphisms of a finite abelian p-group

First, let $A$ be a finitely generated free abelian group, and $s$ an automorphism of order $2$ of $A$. Set $G=\{1,s\}$. Then we know that $A$ is a sum of indecomposable $G$-lattices $A_i$, where ...
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### Torsion in profinite groups

Is there a finitely generated profinite group $G$ with a closed subgroup of infinite index $K \leq G$ such that for every $g \in G$ there exists some $n \in \mathbb{N}$ for which $g^n \in K$ ? Can $G$...
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### A class 3 group of order 243

Let G be a group of order $243=3^5$. We denote by $(G_i)$ its lower central series and assume that $G$ has class $3$ and that $|G:G_2|=|G_3|=9$. We assume moreover that the cubing map factors as a (...
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### Is the Frattini subgroup of a f.g virtually pro-p group open?

Let $G$ be a finitely generated profinite group, and $p$ a prime number. Suppose that there exists some open pro-$p$ subgroup $H \leq_o G$. Must $G$ have only finitely many maximal open subgroups? ...
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### Finite p-groups and their fibered products

Is every finite $p$-group an epimorphic image of a fibered product of two finite $p$-groups which can be generated by $2$ elements?
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### An example of a simple infinite 2-group

I've asked this question before on Mathematics, and they suggested me to ask here (Link). Is there an example of a simple infinite $2$-group? Informations If a $2$-group is Artinian I know that ...
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### Union of conjugates in p-groups

Fix a prime number $p$. Is there a sequence $\{a_k\}_{k \in \mathbb{N}}$ of real numbers with $$\lim_{k \to \infty} a_k = 0$$ such that for any finite $p$-group $G$, and any subgroup $H \leq G$ with ...
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### Exhausting a free pro-p group

Recall that for a profinite group $G$ we define the subgroup rank to be $$\sup \{d(H): H \leq_c G\}$$ where $d(H)$ stands for the minimal cardinality of a set of topological generators of $H$. Let $p$...
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### On a problem of Berkovich

What is the real history of the following problem proposed by Berkovich [Y. Berkovich, Z.Janko, Groups of prime power order. Volume 2, Expositions in Mathematics, 56, Walter de Gruyter, New York, 2011]...
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### An epimorphism into a profinite group

Let $p$ be an odd prime number, $G$ a finitely generated nonabelian profinite group, $L \lhd_o G$ a pro-$p$ group with $[G : L] = 2$. Suppose that there is a continuous surjection from $G$ onto a free ...
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### Variety of nilpotent Lie algebras or $p$-groups

Here's a couple of analogous questions, one in terms of finite-dimensional complex Lie algebras and one in terms of finite $p$-groups; I'd be interested in an answer to either: 1) Let $\mathcal{L}$ ...
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Let $p$ be a prime number, $F$ a free nonabelian finitely generated pro-$p$ group, $L \lhd_o F$ and $Y$ a basis for $L$ with $y \in Y$. Is there a basis $X$ for $F$ such that $y$ is in the abstract ...
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### Dense free subgroups

Let $F$ be a free pro-$p$ group (for a prime number $p$) on a finite set $X$, $\Phi$ the abstract subgroup generated by $X$, $\{1\} \neq N \lhd_c F$. Is it possible that $\Phi \cap N = \{1\}$?
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### Lower central series in a free pro-p group

Let $F$ be a nonabelian finitely generated free pro-$p$ group, $H \leq_c F$ of infinite index. Denote by $\{F_n\}_{n \in \mathbb{N}}$ the lower central series of $F$, and set $r_n = [F : F_nH]$. Is ...
Let $p$ be a prime number, $F$ a free pro-$p$ group, $H \leq_c F$ of infinite index. Can it be that $$\sup_{N \lhd_o F} |Z((F/N)/C_{F/N}(HN/N))| < \infty ?$$
Let $G$ be a finite group, $H \leq G$ a proper subgroup. It is well known that the union of the conjugates of $H$ does not cover $G$. I would like to know of more precise results (even in special ...