Thompson's lemma on product with centralizer of commutator with abelian subgroup of maximum order
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Contents
History
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.
Statement
Suppose is a group of prime power order. Let denote the set of abelian subgroups of maximum order in . Suppose . Suppose is such that the commutator is an abelian subgroup of . Let . Then, .
The operation sending to the new subgroup is termed (on this wiki) the Thompson replacement operation.
Related facts
Applications
- Thompson's replacement theorem for abelian subgroups
- Thompson's replacement theorem for elementary abelian subgroups
Facts used
- 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
Proof
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: .
Proof:
- 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 .
References
Textbook references
- Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 272, Theorem 2.4, Section 8.2 (Glauberman's theorem), ^{More info}
Journal references
- 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 copy}^{More info}