Find the number of elements that (individually) generate \(\mathbb{Z}/49000\mathbb{Z}\).

Let \(\sigma=(1\ 2\ 3\ 4\ 5\ 6\ 7\ 8\ 9\ 10\ 11\ 12)\). For each of the following integers \(a\), compute \(\sigma^a\) for \(a=13\), \(65\), \(626\), \(1195\), \(-6\), \(-81\), \(-570\), and \(-1211\).

Show that if \(|G|=pq\) for some primes \(p\) and \(q\) (not necessarily distinct) then either \(G\) is abelian or \(Z(G)=1\).

Prove that if \(N\) is a normal subgroup of the finite group \(G\), and \(|N|\) and \(|G:N|\) are relatively prime, then \(N\) is the unique subgroup of \(G\) of order \(|N|\).

Prove that if \(H\) is a normal subgroup of \(G\) of prime index \(p\) then for all \(K \le G\) either

• \(K \le H\), or
• \(G=HK\) and \(|K:K \cap H|=p\).

Let \(M\) and \(N\) be normal subgroups of \(G\) such that \(G=MN\). Prove that \(G/(M \cap N) \cong (G/M) \times (G/N)\).

Prove that the unique subgroup of order \(4\) in \(A_4\) is normal and is isomorphic to the Klein \(4\)-group.

Prove that \(A_n\) contains a subgroup isomorphic to \(S_{n-2}\) for every \(n \ge 3\).

Prove that if a group \(G\) acts transitively on a non-empty set \(A\) then the kernel of the action is \[\bigcap_{g\in G} g S g^{-1} \] where \(S\) is the stabilizer of \(a \in A\). (In particular, it is independent of the choice of \(a \in A\).)