Signalizer functor

This article defines a particular kind of map (functor) from a set of elements in the group to a set of subgroups

This article is about a term related to the Classification of finite simple groups

Definition

Definition with symbols

Let $G$ be a group and $A$ an Abelian subgroup of $G$. Then, a signalizer functor on $A$ in $G$ is a map $\theta$ from the set of non-identity elements of $A$ to the set of $A$-invariant subgroups (that is, the set of subgroups of $G$ that commute with every element of $A$).

Properties

Completeness

A signalizer functor is said to be complete if there exists a subgroup $\theta (G)$ such that for every $a$ in $A$, $\theta (a)$ is the same as the centralizer of $a$ in $\theta (G)$.

Solvability

A signalizer functor is said to be solvable if all its images are solvable groups, viz $\theta$ is a solvable signalizer functor if for every $a$, $\theta (a)$ is a solvable group.

Facts

Signalizer functor theorem

Further information: signalizer functor theorem

If $m(A)$ is at least 3, every signalizer functor on $A$ is complete. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

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A special case of this is the solvable signalizer functor theorem.