# Questions tagged [p-groups]

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101
questions

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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 $G$ ...

**3**

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**1**answer

173 views

### 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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**1**answer

79 views

### 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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58 views

### When is the following preorder on the set of central elements of order 2 a total preorder?

Let $G$ be a finite 2-group. Denote by $S$ the set of central elements of $G$ of order exactly $2$. The relation $a\leq b$ iff there is an endomorphism of $G$ sending $b$ to $a$ defines a preorder on $...

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78 views

### A question to the derived length in modular group algebras

Let $p$ be a prime number, $G=:G_1$ a (non-Abelian) finite p-Group and $K$ a finite field with $\operatorname{char}(K)=p$. It is well-known that the group $G_2:=1+\operatorname{rad}(KG)$ is a p-group ...

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15 views

### Exponents in unit groups of modular group algebras

Let $p$ be a prime number, $G=:G_1$ a (non-Abelian) finite p-Group and $K$ a finite field with $\operatorname{char}(K)=p$. It is well-known that the group $G_2:=1+\operatorname{rad}(KG)$ is a p-group ...

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54 views

### Number of conjugacy classes of unit groups of modular group algebras

Let $n$ be a natural number, $p$ a prime number, $G$ a finite $p$-group and $K$ a finite field with $p^n$ elements. We focus on the group $1+J(KG)$, where $J(KG)$ is the Jacobson radical of $KG$, ...

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63 views

### p-groups embedded into Sylow subgroups

Let $p$ be a prime number, $q$ a power of $p$ and $P$ be a finite $p$-group. $P$ is isomorphic to a subroup of p-Sylow subgroup of
the symmetric group $S_{\mid P\mid}$ (Theorem of Cayley)
the ...

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151 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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**1**answer

110 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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139 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(...

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**1**answer

136 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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136 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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**1**answer

123 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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272 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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28 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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155 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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**1**answer

223 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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69 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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139 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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**1**answer

168 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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124 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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143 views

### 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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**1**answer

145 views

### Number of subgroups of a group of orders $p^3$ [closed]

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\...

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175 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 ...

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24 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 $\Omega(P) \subset Z(P)$).
...

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**2**answers

530 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 ...

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**1**answer

599 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 $q\neq p$.
...

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**1**answer

484 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 ...

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228 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 ...

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180 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 $x\in M$, $C_G(x)\subseteq M$. ...

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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 ...

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**2**answers

196 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 ...

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90 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 \...

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**1**answer

263 views

### p-groups such that the center is contained in many cyclic subgroups

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 ...

**7**

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**1**answer

345 views

### 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 ...

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**1**answer

213 views

### Representations of p-groups where 1 is never an eigenvalue

Fix some $n \geq 1$ and some prime $p$. I'm looking for finite $p$-groups $G$ and finite-dimensional complex representations $V$ of $G$ with the following two properties:
The abelianization of $G$ ...

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**1**answer

164 views

### Uniform pro-p groups as a semi-direct product

Let $G$ a finitely generated uniform pro-$p$ group. Then $G/[G,G]$ is abelian and so it is of the form $\mathbb{Z}_p^r\times T$ for some integer $r$ and finite $p$-group $T$. Therefore, $[G,G]$ is ...

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**1**answer

461 views

### p-groups with unique normal minimal subgroup

Have $p$-groups with a unique normal minimal subgroup been classified? Is there any article on the subject?

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188 views

### Maximal cyclic quotient of a $p$-group

Let $G$ be a finite abelian $p$-group, $p$ a prime. I say that a pair $(G',\varphi)$ is a maximal cyclic quotient (please excuse me if this definition already exists and refers to a different concept) ...

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166 views

### 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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175 views

### 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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263 views

### 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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286 views

### 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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3k views

### classification of $p$-groups

I have two questions regarding to $p$-groups.
A $p$-group $G$ is said to be extraspecial of $G'=Z(G)$ has order $p$. Hence extraspecial groups are examples of $p$-groups with cyclic center. Of ...

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**1**answer

205 views

### Finite solvable groups are generated by a nilpotent subgroup + K elements?

Is there a constant $K \in \mathbb{N}$ such that for every finite solvable group $G$, there exists a nilpotent subgroup $N \leq G$, and a subset $S \subseteq G$ with $|S| \leq K$, and $\langle N,S\...

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**1**answer

188 views

### 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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143 views

### finite p-group subgroup of infinite p-group

is there any finite p-group G that is subgroup or minimal/maximal subgroup of infinite p-group H?
if yes what is the limits? can this happen with different p's?
i'm more interested in being maximal ...

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156 views

### Is there a Noetherian profinite group of infinite rank?

Is there a profinite group $G$ such that any closed subgroup $H \leq G$ is finitely generated, but there is no $n \in \mathbb{N}$ such that every closed subgroup of $G$ can be generated by at most $n$ ...

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**1**answer

479 views

### 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 $G$...