# Thompson order formula

In mathematical finite group theory, the **Thompson order formula**, introduced by John Griggs Thompson Template:Harv, gives a formula for the order of a finite group in terms of the centralizers of involutions, extending the results of Template:Harvtxt.

## Statement

If a finite group *G* has exactly two conjugacy classes of involutions with representatives *t* and *z*, then the Thompson order formula Template:Harv Template:Harv states

- |G| = |C
_{G}(*z*)|*a*(*t*) + |C_{G}(*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*.

Template:Harvtxt gives the following more complicated version of the Thompson order formula for the case when *G* has more than two conjugacy classes of involution.

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

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*.

## References

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