Central factor

Terminological note
The term central factor used here refers to a subgroup that can occur as a factor in a central product. The term central factor is also used in another, completely different sense: a group that occurs as a quotient by a central subgroup in another group, or in some cases, as a quotient by the whole center (in which case, it would be isomorphic to the inner automorphism group). These other senses of the word are very different, and you can learn more about these senses at the central extension page.

Extreme examples

 * The trivial subgroup in any group is a central factor.
 * Every group is a central factor in itself.

Analogues in other algebraic structures
For a complete list, refer:

(generated semantically).

Conjunction with other properties
Some conjunctions with group properties:


 * Central subgroup is a central factor that is also an abelian group.
 * Nilpotent central factor is a central factor that is also a nilpotent group.

Some conjunctions with subgroup properties:


 * Weaker than::Characteristic central factor: A central factor that is also a characteristic subgroup.

Formalisms
The subgroup property of being a central factor has a first-order description as follows. A subgroup $$H$$ is a central factor in a group $$G$$ if and only if:

$$\forall g \in G, \exists h \in H. \forall x \in H, \qquad hxh^{-1} = gxg^{-1}$$

This is a Fraisse rank 2 expression.

In particular, this means that the property of being a central factor is a satisfies metaproperty::left-inner subgroup property.

A subgroup $$H$$ of a group $$G$$ is a central factor if and only if the following is true: there exists a group $$K$$, a direct factor $$L$$ of $$K$$, and a surjective homomorphism $$\rho:K \to G$$ such that $$\rho(L) = H$$.

A subgroup $$H$$ of a group $$G$$ is a central factor if and only if there exist groups $$K_i, i \in I$$ all contained in $$G$$ such that $$H$$ is a cocentral subgroup of each $$K_i$$ (i.e., $$HZ(K_i) = K_i$$) and the join of the $$K_i$$s equals $$G$$.