Show that if \(G/Z(G)\) is cyclic then \(G\) is abelian.
Show that \(G/N\) is abelian iff \([G, G] \leq N\).
Show that normal subgroups are not necessarily contained in \(Z(G)\).
- Hint: consider the order 3 subgroup of \(S_3\).
Solution
If \(G/Z(G)\) is cyclic, then \(G\) is abelian:
Write \(H\da Z(G)\) and \(G/H = \gens{xH}\) as a cyclic quotient.
Fix \(a, b\in G\), then \(aH = x^n H\) and \(bH = x^m H\).
So \(ax^{-n} = h_1, bx^{-m} = h_2\) where the \(h_i\) are now central.
Now write \(ab = (x^n h_1)(x^m h_2) = ba\) by commuting everything.