91Ó°ÊÓ

Show that if every element of a group \(G\) is equal to its own inverse, then \(G\) is Abelian.

Let \(H\) and \(K\) be cyclic subgroups of an Abelian group \(G,\) with \(|H|=10\) and \(|K|=14\). Show that \(G\) contains a cyclic subgroup of order 70 .

Let \(G\) be a nonempty finite set closed under an associative operation such that both the left and the right cancellation laws hold. Show that \(G\) under this operation is a group.

Find all the elements in \(S_{4}\) of order \(2 .\)

Show that if \(\sigma \in A_{n}\), then \(\sigma\) can be written as a product of 3 -cycles.