## Characteristic subgroups and its properties

A subgroup $latex H$ of a group $latex G$ is called characteristic in $latex G$, denoted $latex H$ char $latex G$, if every automorphism of $latex G$ maps $latex H$ to itself, i.e., $latex \sigma (H) = H$ for all $latex \sigma \in Aut(G)$.

Results concerning characteristic subgroups which we shall use later (and whose proofs are relegated to the exercises) are

(1) characteristic subgroups are normal,

(2) if $latex H$ is the unique subgroup of $latex G$ of a given order, then $latex H$ is characteristic in $latex G$, and

(3) if $latex K$ char $latex H$ and $latex H \trianglelefteq G$, then $latex K$ $latex \trianglelefteq G$ (so although “normality” is not a transitive property (i.e., a normal subgroup of a normal subgroup need not be normal), a: characteristic subgroup of a normal subgroup is normal).

(4) the relation of “being characteristic” is transitive.

For proofs click