Let be a subgroup of a group . The similarity transformation of by a fixed element in not in always gives a subgroup. If
for every element in , then is said to be a normal subgroup of , written (Arfken 1985, p. 242; Scott 1987, p. 25). Normal subgroups are also known as invariant subgroups or self-conjugate subgroup (Arfken 1985, p. 242).
All subgroups of Abelian groups are normal (Arfken 1985, p. 242).