All Questions
Tagged with gr.group-theory p-groups
121 questions
4
votes
2
answers
832
views
p-group with abelian centralizer
I will be so thankful if someone helps me with the following question. There exists finite non-abelian p-groups G (except non-abelian groups of order $p^3$) with the following properties:
all non-...
5
votes
3
answers
581
views
Normal abelian subgroups in p-groups
Given a group $G$, we denote by $T(G)$ the subgroup generated by all (maximal) normal abelian subgroups of $G$.
Let define the series $(T_i(G))$ by $T_0(G)=1$ and $T_{i+1}(G)/T_i(G)=T(G/T_i(G)$, and $...
2
votes
2
answers
290
views
P-group with abelian centralzer
I will be so thankful if someone help me about the following question.
I need to know the presentation of a (if it is possible) family of finite non-abelian $p$-group $G$ with the follwing properties:
...
- 139
7
votes
3
answers
923
views
Characters of p-groups
Berkovich mentioned the following result of Mann in his book on p-groups:
The number of nonlinear irreducible characters of given degree in a p-group is divided by p-1.
Do you know any reference for ...
- 307
1
vote
1
answer
242
views
Normal subgroups In a p-group [Reference?]
Dear Experts,
I'm a graduate student, dealing with group-theory.
In my current research, I used the bound "Alexander Gruber" wrote about in this post:
See Here
(Actually, I have just found out ...
- 35
4
votes
1
answer
421
views
Generators of p-groups
Let $G$ be a finite $p$-group. Since we can embed $Z_2(G)/Z(G)$ in $Hom(G,Z(G))$, we have $d_2 \leq d(G)d(Z(G))$; where $d_2(G)=d(Z_2(G)/Z(G))$ and $d(G)$ denotes the minimal number of generators of $...
7
votes
3
answers
627
views
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 $M(p^n)$)
- 139
33
votes
2
answers
1k
views
Richness of the subgroup structure of p-groups
Given a prime $p$ and $n \in \mathbb{N}$, let $f_p(n)$ be the smallest
number such that there is a group of order $p^{f_p(n)}$ which all groups of
order $p^n$ embed into. What is the asymptotic growth ...
- 19.6k
1
vote
2
answers
602
views
finite abelian p-groups with solvable automorphism group
Let $G$ be an abelian (not elementary) finite $p$-group. In what conditions the automorphism group of $G$ is solvable?
- 792
10
votes
5
answers
980
views
Automorphism Group of a p-group : Looking for a Reference
In the following post by DavidLHarden :
See Here
He quoted the following claim:
"There is a theorem that says that if $p$ is a prime and $|G|=p^n $ , then $|AutG| $ divides
$ \Pi_{k=0}^{n-1} (p^{n}-...
- 253
10
votes
3
answers
6k
views
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 ...
- 253
5
votes
3
answers
384
views
Hall algebra for non-abelian $p$-groups?
According to WP article on Hall algebras one counts the number of abelian subgroups in an abelian group with fixed type of subgroup, group, quotient.
Two things are claimed:
These numbers are ...
- 24.9k
6
votes
4
answers
4k
views
Center of p-groups
Is it true that any abelian $p$-group (not necessarily finite) is the center of a $p$-group and of index $p^2$?
4
votes
3
answers
2k
views
Representation theory of p-groups in particular upper tringular matrices over F_p
Finite p-groups - have p^n elements by definition. According to WP there is rich structure theory.
Question: How far is representation theory of p-groups is understood?
In case this question is too ...
- 24.9k
10
votes
1
answer
586
views
Maximal subgroups of a certain finite 2-group
The following came up in a problem on reconstruction of digraphs. I determined enough about the answer to satisfy the application completely, but still I am curious to know what the complete solution ...
- 37.7k
8
votes
2
answers
2k
views
Representation theory of a finite p-group over a field of characteristic p: dim of invariants =1 => dim of coinvariants = 1?
I am trying to understand the proof of Proposition 4 in
S. Ullom, Integral normal bases in Galois extensions of local fields, Nagoya Math. J. Volume 39 (1970), 141-148. The PDF is available here:
http:...
- 1,661
-1
votes
1
answer
469
views
Why every $p$-group of infinite order is not simple?
Why every $p$-group of infinite order is not simple?
- 15
4
votes
4
answers
1k
views
Maximum value of the number of conjugacy classes of nonabelian p-groups with an abelian subgroup of index p
It is known that if $G$ is a nonabelian $p$-group of order $p^n$, with an abelian subgroup of index $p$, then the number $k(G)$ of conjugacy classes of $G$ can be as large as $p^{n-1} + p^{n-2} - p^{n-...
- 49
5
votes
2
answers
761
views
Center of finite metabelian p-groups
$\DeclareMathOperator\rk{rk}$
Let $G$ be a finite metabelian $p$-group, i.e. the commutator subgroup $G'$ of $G$ is abelian. Then I ask myself under which conditions does the following hold:
$$\tag{$*...
4
votes
3
answers
502
views
Molien for modular representations?
Let $G$ be a finite group, and let $k$ be a field whose characteristic divides $\left|G\right|$. Let $\rho:G\to \mathrm{End} V$ be a (finite-dimensional) representation of $G$ over $k$. Prove or ...
- 33.8k
10
votes
3
answers
956
views
faithful unipotent representations of (finite) $p$-groups
The title pretty much summarizes the question: does every $p$-group have a faithful unipotent representation (with coefficients in $\mathbb{F}_p$ or some finite extension thereof)?
- 96.4k