Question:Normal subgroup center relation
This question is about normal subgroup and center| See more questions about normal subgroup | See more questions about center
This question has type relation between two concepts
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.
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).