prove that the center of a group is a normal subgroup?

Answer 1

Call the center Z

Z={g in G : gh = hg for all h in G} is the definition of center

goal: show gZ = Zg for all g in G where gZ = {gz : z in Z}

but every element gz is in Zg since gz = zg
and every element zg is in gZ for the same reason. So gZ = Zg

EDIT: The answer below is wrong by the way. It should be that ghg^-1 = some element of H (doesnt have to be h)!

Answer 2

For a subgroup H to be a normal subgroup of a group G
we must have ghg^-1 = h for g in G and h in H.
But h commutes with everything in G, so
ghg^-1 = hgg^-1 = h,
so we are done.