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In the mathematical classification of finite simple groups, the component theorem of Aschbacher (1975, 1976) shows that if G is a simple group of odd type, and various other assumptions are satisfied, then G has a centralizer of an involution with a "standard component" with small centralizer.

References

Aschbacher, Michael (1975), "On finite groups of component type", Illinois Journal of Mathematics 19: 87–115, ISSN 0019-2082, MR 0376843
Aschbacher, Michael (1976), "Tightly embedded subgroups of finite groups", Journal of Algebra 42 (1): 85–101, doi:10.1016/0021-8693(76)90028-4, ISSN 0021-8693, MR 0422400

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