s is an additive subgroup: a set containing 0 and closed under addition and negation.

s is a subgroup: a set containing 1 and closed under multiplication and inverse.

The trivial subgroup

Every subgroup is a normal subgroup of its normalizer

group.closure s is the subgroup closed over s, i.e. the smallest subgroup containg s.

Given an element a, conjugates a is the set of conjugates.

Given a set s, conjugates_of_set s is the set of all conjugates of the elements of s.

The set of conjugates of s is closed under conjugation.

The normal closure of a set s is the subgroup closure of all the conjugates of elements of s. It is the smallest normal subgroup containing s.

The normal closure of a set is a subgroup.

The normal closure of s is a normal subgroup.

The normal closure of s is the smallest normal subgroup containing s.