Thompson replacement operation

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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 $P$ is a group of prime power order. Let $\mathcal{A}(P)$ denote the set of abelian subgroups of maximum order in $P$. Suppose $A \in \mathcal{A}(P)$. Suppose $x \in P$ is such that the commutator $[x,A]$ is an abelian subgroup $M$ of $P$.

The Thompson replacement operation for $x$ is defined as:

$A \mapsto MC_A(M)$.

The Thompson replacement operation can roughly be thought of as a way of moving from a less normal element of $\mathcal{A}(P)$ to a more normal element of $\mathcal{A}(P)$.

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 $x \in N_P(A)$, then the Thompson replacement operation for $x$ sends $A$ to itself. In particular, if $A$ is normal in $P$, then it is fixed by the Thompson replacement operation for every element of $P$.
• 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 $p^5$. This is because for groups of order up to $p^4$, every abelian subgroup of maximum order is normal.