# Thompson order formula

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In mathematical finite group theory, the Thompson order formula, introduced by John Griggs Thompson ( Held 1969 , p.279), gives a formula for the order of a finite group in terms of the centralizers of involutions, extending the results of Brauer & Fowler (1955).

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In group theory, a branch of mathematics, the term order is used in two unrelated senses:

## Statement

If a finite group G has exactly two conjugacy classes of involutions with representatives t and z, then the Thompson order formula ( Aschbacher 2000 , 45.6)( Suzuki 1986 , 5.1.7) states

|G| = |CG(z)|a(t) + |CG(t)|a(z)

Here a(x) is the number of pairs (u,v) with u conjugate to t, v conjugate to z, and x in the subgroup generated by uv.

Harris (1972 , 3.10) gives the following more complicated version of the Thompson order formula for the case when G has more than two conjugacy classes of involution.

${\displaystyle |G|=C_{G}(t)C_{G}(z)\sum _{x}{\frac {a(x)}{C_{G}(x)}}}$

where t and z are non-conjugate involutions, the sum is over a set of representatives x for the conjugacy classes of involutions, and a(x) is the number of ordered pairs of involutions u,v such that u is conjugate to t, v is conjugate to z, and x is the involution in the subgroup generated by tz.

## Proof

The Thompson order formula can be rewritten as

${\displaystyle {\frac {|G|}{C_{G}(z)}}{\frac {|G|}{C_{G}(t)}}=\sum _{x}a(x){\frac {|G|}{C_{G}(x)}}}$

where as before the sum is over a set of representatives x for the classes of involutions. The left hand side is the number of pairs on involutions (u,v) with u conjugate to t, v conjugate to z. The right hand side counts these pairs in classes, depending the class of the involution in the cyclic group generated by uv. The key point is that uv has even order (as if it had odd order then u and v would be conjugate) and so the group it generates contains a unique involution x.

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## References

Michael George Aschbacher is an American mathematician best known for his work on finite groups. He was a leading figure in the completion of the classification of finite simple groups in the 1970s and 1980s. It later turned out that the classification was incomplete, because the case of quasithin groups had not been finished. This gap was fixed by Aschbacher and Stephen D. Smith in 2004, in a pair of books comprising about 1300 pages. Aschbacher is currently the Shaler Arthur Hanisch Professor of Mathematics at the California Institute of Technology.

Cambridge University Press (CUP) is the publishing business of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the world's oldest publishing house and the second-largest university press in the world. It also holds letters patent as the Queen's Printer.

The International Standard Book Number (ISBN) is a numeric commercial book identifier which is intended to be unique. Publishers purchase ISBNs from an affiliate of the International ISBN Agency.