91Ó°ÊÓ

(i) Prove that the special orthogonal group \(S O(2, \mathbb{R})\) is isomorphic to the circle group \(S^{1}\). (ii) Prove that all the rotations of the plane about the origin form a group under composition that is isomorphic to \(\mathrm{SO}_{2}(\mathrm{R})\).

Let \(H\) and \(K\) be subgroups of a group \(G\). (i) Prove that \(H K\) is a subgroup of \(G\) if and only if \(H K=K H\). In particular, the condition holds if \(h k=k h\) for all \(h \in H\) and \(k \in K\). (ii) If \(H K=K H\) and \(H \cap K=\\{1\\}\), prove that \(H K \cong H \times K\).

(i) How many permutations in \(S_{5}\) commute with (1 2)(3 4), and how many even permutations in \(S_{5}\) commute with (1 2)(3 4). (ii) How many permutations in \(S_{7}\) commute with (1 2) (3 4 5)? (iii) Exhibit all the permutations in \(S_{7}\) commuting with (1 2)(3 4 5).

(i) Show that there are two conjugacy classes of 5 -cycles in \(A_{5}\), each of which has 12 elements. (ii) Prove that the conjugacy classes in \(A_{5}\) have sizes \(1,12,12,15\), and \(20 .\)

Let \(G\) be a finite group written multiplicatively. Prove that if \(|G|\) is odd, then every \(x \in G\) has a unique square root, that is, there exists exactly one \(g \in G\) with \(g^{2}=x\).

Prove that \(A_{5}\) is a group of order 60 that has no subgroup of order 30 .

Prove that \(A_{4}\) is the only subgroup of \(S_{4}\) of order 12. (In Exercise 2.123 on page 205 , this will be generalized from \(S_{4}\) to \(S_{n}\) for all \(n \geq 3\).)

Prove that the symmetry group \(\Sigma\left(\pi_{n}\right)\), where \(\pi_{n}\) is a regular polygon with \(n\) vertices, is isomorphic to a subgroup of \(S_{n}\).