Commutator subgroup and center

[ehrenfest]

Homework Statement

Please confirm that the center of a group always contains the commutator subgroup. I am pretty sure its true.

Homework Equations

The Attempt at a Solution

 

[StatusX]

It's not. That would imply commutators commute, which there's no good reason to expect, and it should be easy to find a counterexample.

 

[ehrenfest]

But my book has a theorem that says:

"If N is a normal subgroup of G, then G/N is abelian if and only if C is contained in N"
where C is the commutator subgroup.

Clearly the center is normal and its quotient group is abelian!

 

[morphism]

G/Z(G) is not necessarily abelian, e.g. Z(A_5) is trivial (since A_5 is simple), whence A_5/Z(A_5) =~ A_5, a non-abelian group.