Centers

The center of a group is the set of group elements which commute with every element in the group. Not to be confused with the commutators, which make two elements commute, but needn't themselves commute with anything. A central element c obviously obeys gc = cg for all group elements g. The center is a subgroup: gc₁c₂ = c₁gc₂ = c₁c₂g, so the product of two central elements is itself a central element. And the inverse of a central element also commutes with everything:

gc^-1 = h

g = hc

g = ch

c^-1g = h = gc^-1

The center is another example of a fully invariant subgroup, as commutativity is preserved by a homomorphism.

If a group is commutatative, the center is the whole group (all elements commute with everything) and the commutator subgroup is $\{I\}$. Otherwise the center will be a proper subgroup (possibly) $\{I\}$ and the commutator subgroup will be a non-trivial subgroup (and possibly the whole group.)