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2h
revised Subgroups from which all class functions extend to class functions on the ambient group
added example.
2h
revised Subgroups from which all class functions extend to class functions on the ambient group
Mentioned fusion/transfer type result. Corrected typos
14h
comment Subgroups from which all class functions extend to class functions on the ambient group
I'm not sure what you would consider a "geometric" reason. There are any number of algebraic expanations, including the fact that $Q_{8}$ admits ${\rm Sp}(2,2) \cong {\rm SL}(2,2)$ as a group of automorphisms.
14h
revised Subgroups from which all class functions extend to class functions on the ambient group
added 110 characters in body
15h
answered Subgroups from which all class functions extend to class functions on the ambient group
22h
revised Properties to have matrices that commute in $\mathrm{GL}_n(\mathbb C)$
Tidying up text
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revised Properties to have matrices that commute in $\mathrm{GL}_n(\mathbb C)$
minor bibliographic correction
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awarded  Enlightened
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awarded  Good Answer
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comment Properties to have matrices that commute in $\mathrm{GL}_n(\mathbb C)$
@Yves: Yes, I was just pointing out that although finiteness is essential for the result as stated, there are things that can be said in some infinite groups- I wasn;t disputing the appropriateness of the "finite groups" tag
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comment Properties to have matrices that commute in $\mathrm{GL}_n(\mathbb C)$
@YvesCornulier: Although there as an important extension of this type of result to discrete subgroups of linear groups by Zassenhaus, which led to the important notion of Zassenhaus neighbourhoods in Lie groups.
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revised Properties to have matrices that commute in $\mathrm{GL}_n(\mathbb C)$
typo
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awarded  Mortarboard
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revised Properties to have matrices that commute in $\mathrm{GL}_n(\mathbb C)$
Rearrangment, some explanation
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revised Properties to have matrices that commute in $\mathrm{GL}_n(\mathbb C)$
Expanded explanations, gave additional references.
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revised Bound for the Frattini subgroup of a $p$-group
deleted 1 character in body
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revised Bound for the Frattini subgroup of a $p$-group
deleted 1 character in body
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awarded  Nice Answer
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revised How to transform matrix to this form by unitary transformation?
corrections
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comment How to transform matrix to this form by unitary transformation?
@Frederik Poloni: I was aware that $UMV$ would not have the same spectrum as $M.$ However, it is true that $UMV$ has the same operator norm (with respect to Euclidean norm on vectors ) as $M,$ namely $m_{1}.$ I was still careless, because this certainly need not imply that $UMV$ has spectral radius $m_{1}.$