Thompson's lemma on product with centralizer of commutator with abelian subgroup of maximum order
This lemma explicitly appears in Gorenstein's text on Finite Groups (see the reference below). It was part of Thompson's proof of his replacement theorem -- however, Thompson did not state it as a separate lemma.
The operation sending to the new subgroup is termed (on this wiki) the Thompson replacement operation.
- Thompson's replacement theorem for abelian subgroups
- Thompson's replacement theorem for elementary abelian subgroups
- Equivalence of definitions of maximal among abelian subgroups: An abelian subgroup of a group that is not contained in any bigger abelian subgroup is a self-centralizing subgroup: it equals its own centralizer.
- Product formula
- Formula for commutator of element and product of two elements
- Witt's identity
Given: A group of prime power order. is the set of abelian subgroups of maximum order in . A subgroup . An element in such that is abelian. .
To prove: .
- is abelian: By assumption, is abelian. Since , is also abelian. Further, every element of commutes with every element of by definition, so is abelian.
- : Since is abelian of maximum order, it is maximal among abelian subgroups: it is not contained in a bigger abelian subgroup. Thus, by fact (1), .
- : Clearly, since , . Conversely, anything in is in particular inside , so centralizes , hence is in . Thus, , and we get . Finally, since step (2) yields , we get .
- : By fact (2), . By step (3), . Thus, we get .
- Consider the map from to . Then, if and are in the same coset of , then : Suppose and are in the same coset of in . Then, by fact (3), . Since the left side is in , so is the right side. By definition of , we get that .
- If and , then : If , then is the identity for all . By fact (4), and the abelianness of , we obtain that is the identity for all . In particular, commutes with , so .
- The map from to has the property that if and are in the same coset of , then are in the same coset of . Thus, it is an injective map from to : This follows from the previous two steps.
- , hence : This follows from the previous step.
- From steps (4) and (6), we get .
- Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 272, Theorem 2.4, Section 8.2 (Glauberman's theorem), More info
- A replacement theorem for p-groups and a conjecture by John Griggs Thompson, Journal of Algebra, ISSN 00218693, Volume 13, Page 149 - 151(Year 1969): In this paper, Thompson proved the replacement theorem for abelian subgroups, as well as the result that group of prime power order having a larger abelianization than any proper subgroup has class two.Official copyMore info