All Questions
10 questions
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
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
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
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
5
votes
1
answer
211
views
The rank of indecomposable finite abelian 2-group
$\DeclareMathOperator\rank{rank}$Let $P$ be a finite $p$-group. The rank of $P$ is $\log_{p}|P/\Phi(P)|$ where $\Phi(P)$ is the Frattini subgroup of $P$, we write $\rank(P)=\log_{p}|P/\Phi(P)|$.
Let a ...
- 613
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
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
4
votes
1
answer
152
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 ...
- 5,654
4
votes
1
answer
360
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 $G$ ...
- 41
1
vote
1
answer
192
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 ...
- 14.1k