20
votes
1answer
731 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 ...
2
votes
1answer
130 views

Hall algebra for non-abelian p-groups ?

According to WP article on Hall algebras one counts the number of abelian subgroups in abelian group with fixed type of subgroup, group, quotient. Two things are claimed: 1) These numbers are ...
3
votes
3answers
444 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 ...
7
votes
2answers
561 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: ...
4
votes
3answers
377 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 ...
8
votes
3answers
664 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)?