Prove that subgroups and quotient groups of a solvable group are solvable.

Prove that if \(K\) is a normal subgroup of \(G\) then \(K'\) is normal in \(G\).

Assume that \(K\) is a cyclic group, \(H\) is an arbitrary group and \(\varphi_1\) and \(\varphi_2\) are homomorphisms from \(K\) into \(\operatorname{Aut}(H)\) such that \(\varphi_1(K)\) and \(\varphi_2(K)\) are conjugate subgroups of \(\operatorname{Aut}(H)\). If \(K\) is infinite assume \(\varphi_1\) and \(\varphi_2\) are injective. Prove that \(H \rtimes_{\varphi_1} K\) is isomorphic to \(H \rtimes_{\varphi_2} K\).

Classify all groups of order \(75\) up to isomorphism.

Establish a finite presentation for \(S_4\) using \(2\) generators.