# Thompson replacement operation

From Groupprops

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This is the replacement operation underlying Thompson's lemma on product with centralizer of commutator with abelian subgroup of maximum order, which in turn is used to prove Thompson's replacement theorem for abelian subgroups.

## Definition

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 .

The **Thompson replacement operation** for is defined as:

.

The Thompson replacement operation can roughly be thought of as a way of moving from a *less* normal element of to a *more* normal element of .

## Facts

### Directly relevant facts

- By Thompson's lemma on product with centralizer of commutator with abelian subgroup of maximum order, the image of this is also an abelian subgroup of maximum order.
- If , then the Thompson replacement operation for sends to itself. In particular, if is normal in , then it is fixed by the Thompson replacement operation for
*every*element of . - The smallest order for which the Thompson replacement operation results in an actual replacement (i.e., does not send the subgroup to itself) is for a group of order . This is because for groups of order up to , every abelian subgroup of maximum order is normal.