All Questions
69 questions
1
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0
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109
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Center of factors of a finite $p$-group, obtained from a minimal normal subgroup
throughout a research problem about finite $p$-groups,
I have a challenge as follows,
Let $G$ be a finite non-abelian $p$-group, where $p$ is odd and $Z(G)$ is non-cyclic.
($Z(G)$ denotes the center ...
- 111
1
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0
answers
105
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Groups $P$ of order $p^5$ with $\Omega_1(P)=P$
I have been working with (particular) groups $P$ of order $p^5$. In fact, the ones that interest me the most are those that satisfy $$\langle x\in P\mid x^p=1\rangle=:\Omega_1(P)=P.$$ After a search ...
- 245
1
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0
answers
128
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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 ...
- 381
1
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0
answers
88
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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-...
- 381
1
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0
answers
65
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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 $...
user145520
1
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0
answers
67
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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 $KG$, ...
- 581
1
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0
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193
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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(...
- 41
1
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0
answers
168
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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 ...
- 405
1
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0
answers
145
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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 ...
- 487
1
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0
answers
239
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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 ...
- 487
1
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0
answers
134
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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 $\Phi \cap N = \{1\}$?
- 11.3k
1
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0
answers
166
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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 ...
- 11.3k
1
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0
answers
82
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A bound on the size of the center
Let $p$ be a prime number, $F$ a free pro-$p$ group, $H \leq_c F$ of infinite index. Can it be that $$\sup_{N \lhd_o F} |Z((F/N)/C_{F/N}(HN/N))| < \infty ?$$
- 11.3k
1
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1
answer
308
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Subgroups of the union of conjugates
This question is an attempt to find a version of this question with a positive answer. In this question, we ask if one can find big subgroups in the union of conjugtes of small subgroups of finite $p$-...
- 11.3k
0
votes
2
answers
956
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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 ...
- 487
0
votes
1
answer
198
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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 $x\in M$, $C_G(x)\subseteq M$. ...
- 71
0
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0
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96
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The relation between two characteristic subgroups in finite p-group
Suppose $G$ is a finite $p$-group. Let
\begin{align*}
\mho_{1}(G)=\langle a^p\mid a\in G\rangle,\quad\Omega_{1}(G)=\langle a\in G\mid a^p=1\rangle.
\end{align*}
There are examples such that $|G|\leq |\...
- 171
0
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0
answers
345
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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(...
- 415
-2
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1
answer
162
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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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