Question:Normal subgroup center relation

Q: '''I learned something about normal subgroup and also about something called the center. What is the difference between the notion of center and the notion of normal subgroup?'''

A: The center is defined as the set of elements that are, as individual elements, invariant under conjugation by other elements. On the other hand, a normal subgroup has to be invariant under conjugation as a set -- conjugation may move elements within the set.

It is true that the center is normal. More generally, a central subgroup is a subgroup of the center, and any central subgroup is normal.

However, every normal subgroup need not be central. In fact, even an abelian normal subgroup need not be central. Examples include the normal subgroup of order three in the symmetric group of degree three (see subgroup structure of symmetric group:S3 and A3 in S3) and the normal subgroups of order four in the dihedral group:D8 (see subgroup structure of dihedral group:D8, cyclic maximal subgroup of dihedral group:D8, Klein four-subgroups of dihedral group:D8, and index two implies normal).