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In the question Math.SE #16716, Natalia asked about representing rings of matrices as centralizers of a matrix. This is an intriguing question, but had some clear problems as rings of matrices need not contain their own centralizers, which would be pretty bad.

In finite groups, subgroups that contain their own centralizers have an eery importance. It occurred to me that subalgebras of matrix algebras probably have some similar properties, but I don't know what to search for.

What do you call a subalgebra that contains its own centralizer?

In other words, is there a name for a collection of matrices:

  • that contains the scalar matrices,
  • that is closed under addition and multiplication, and
  • that contains every matrix that commutes with every matrix in the collection?
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As in the case of groups, the term "self-centralizing subalgebra" seems to be used by some people. – Chris Eagle Jan 8 '11 at 1:02
Is there a connotation of being commutative? It is often, but not always, the case for finite groups that "a self-centralizing subgroup" must not only contain its centralizer, but also "centralize itself". Whether or not the definition requires it, the usage (google hits) seems to be predominantly for self-centralizing abelian sub-thingies. I'd like analogues of words like "centric" (possibly non-abelian self-centralizing subgroups of p-groups) or "constrained" (possibly non-abelian self-centralizing normal p-subgroups). – Jack Schmidt Jan 8 '11 at 2:22
Ah, I did find Glauberman using "self-centralizing" for a non-abelian subgroup and by other authors in follow-ups, so it is de facto a "standard use". I'd still like a few more non-abelian sounding words to help with searches. Feel free to post it as answer, as it is the best I've seen so far. :-) – Jack Schmidt Jan 8 '11 at 2:31
@ChrisEagle Consider posting as an answer, please? Thank you! – rschwieb Oct 7 '14 at 12:48

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