Characteristic subgroup

In the group theory is a characteristic subgroup of a group , a subgroup , which under each automorphism of is mapped into itself. ${\displaystyle G}$ ${\displaystyle H}$ ${\displaystyle G}$

definition

A subgroup of is called characteristic if for every automorphism, that is, bijective group homomorphism , always holds. ${\displaystyle H}$ ${\displaystyle G}$ ${\displaystyle f\colon G\rightarrow G}$ ${\displaystyle f(H)\subset H}$

properties

Every characteristic subgroup is a normal subgroup , because it is preserved especially under every internal automorphism . Conversely, however, not every normal factor is characteristic. Consider e.g. B. the Klein group of four . Each of their subgroups is normal, but there is an automorphism that permutes the 2-element subgroups, so none of the 2-element subgroups is characteristic.

The group itself and the trivial subgroup , which only consists of the neutral element , are always characteristic. If there are no other characteristic subgroups, the group is called characteristically simple , Klein's group of four is an example after what has just been said.

If the finite group is a normal divisor and has no further subgroups of the same order, then it is characteristic that automorphisms only map subgroups to subgroups with the same order . ${\displaystyle H}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle H}$

Strictly characteristic subgroup

A related concept is that of a distinguished subgroup . Such a subgroup remains fixed under each epimorphism ( surjective homomorphism) from to . Note that for an infinite group, not every epimorphism has to be an automorphism. For finite groups, however, the terms characteristic subgroup and strictly characteristic subgroup coincide. ${\displaystyle H}$ ${\displaystyle G}$ ${\displaystyle G}$

Fully characteristic subgroup

→ Main article : Fully invariant subgroup

An even stronger requirement is a fully characteristic or vollinvarianten subgroup (Engl. Fully characteristic subgroup or fully invariant subgroup ). Such a subgroup is mapped into itself under each endomorphism (homomorphism from to ), i. H. if is a homomorphism then is . ${\displaystyle H}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle f\colon G\rightarrow G}$ ${\displaystyle f(H)\subset H}$

Examples

Each fully characteristic subgroup is therefore strictly characteristic, but not the other way around. The center of a group is always strictly characteristic, but z. B. not fully characteristic of the group (the direct product of the dihedral group of order 6 with the cyclic group of order 2). ${\displaystyle D_{6}\times C_{2}}$

The commutator group of a group is always fully characteristic in it, as is the torsional subgroup of an Abelian group .

The property of being characteristic or fully characteristic is transitive ; H. is a (fully) characteristic subgroup of and a (fully) characteristic subgroup of , then is also a (fully) characteristic subgroup of . ${\displaystyle H}$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle G}$ ${\displaystyle H}$ ${\displaystyle G}$

Individual evidence

1. ^ DJS Robinson : A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 0-387-94461-3 , page 28: Characteristic and Fully-Invariant Subgroups