Question

Write 2 characteristic features of the sub groups?

Answer 1

1. a subgroup that is mapped to itself by every automorphism of the parent group. Because every conjugation map is an inner automorphism, every characteristic subgroup is normal; though the converse is not guaranteed.
2. Examples of characteristic subgroups include the commutator subgroup and the center of a group.

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Answer 2

Examples using subgroup-defining functions across all groups

In a non-abelian group, some typical examples of characteristic subgroups are given by subgroup-defining functions (something which uniquely returns a particular subgroup). For instance, we have the following subgroup-defining functions, some of which are interesting for abelian groups as well: