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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 
1d

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 
1d

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 $m_{1}.$ 