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7
votes
2answers
719 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: ...
3
votes
2answers
583 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} - ...
2
votes
4answers
1k views

When a group ring is a local ring [closed]

Hi there, I'm stuck with my undergraduate thesis on the following proposition: If $k$ is a field of characteristic $p > 0$ and $G$ is a finite $p$-group, then the group ring $kG$ is local. In ...
4
votes
2answers
489 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: ...
13
votes
0answers
285 views

p-groups as rational points of unipotent groups

Is it true that every finite p-group can be realized as the group of rational points over $\mathbb{F_p}$ of some connected unipotent algebraic group defined over $\mathbb{F_p}$? For abelian p-groups, ...
4
votes
3answers
391 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 ...
9
votes
3answers
713 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)?