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Results tagged with finite-groups
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user 9672
Questions on group theory which concern finite groups.
0
votes
Reference request: automorphism of abelian $p$-groups of rank 2
Yemon Choi is right to say that for rank two groups this is essentially an exercise. However, a somewhat indirect (and possibly more difficult) approach seems to be instructive here:
Suppose $P$ is a …
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3
votes
Accepted
A question on automorphisms of finite abelian groups
Here is an outline for odd order abelian $p$-groups:
The main point is that every non-zero element in a field with at least three elements is a sum of two non-zero elements. Using this, you can show …
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3
votes
Character Values for Alternating Groups of degree $\geq 7$
This answer essentially summarizes information from the other answers, hopefully, making the whole picture clear. For each self-conjugate partition $\lambda$ (i.e., $\lambda=\lambda'$) of $n$, the irr …
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2
votes
Decomposing representations of GL(n,F_q) induced from certain kinds of parabolics
If $m=1$ and $n>1$, then the decomposition is multiplicity-free and has $q$ irreducible representations. The way to see this is the following: The representation of $GL_n(\mathbf F_q)$ that you are lo …
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2
votes
Decomposition of induced representations / Refinement of Mackey's criterion
The proof of Mackey's theorem on intertwiners actually tells you how to construct the endomorphism algebra of an induced representation, not just its dimension. So, if you work a little harder, you ma …
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5
votes
1
answer
216
views
To whom is the internal characterization of $Q$-groups due?
A group is said to be a $Q$-group if the character of any complex representation is rational valued. A well-known internal characterization of $Q$-groups is the following:
$G$ is a $Q$-group if an …
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13
votes
Automorphism group of a finite group
A nice set of generators for the automorphism group of a finite abelian group is described by Garrett Birkhoff in his paper titled "Subgroups of abelian groups", Proc. London Math. Soc., s2-38(1):385- …
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