91Ó°ÊÓ

(i) Prove that \(S_{n}\) can be generated by \((12),(13), \ldots,(1 n)\). (ii) Prove that \(S_{n}\) can be generated by \((12),(23), \ldots,(i i+1), \ldots,(n-1, n) .\) (iii) Prove that \(S_{n}\) can be generated by the two elements ( 12 ) and \((12 \ldots, n)\). (iv) Prove that \(S_{4}\) cannot be generated by \(\left(\begin{array}{llll}1 & 3\end{array}\right)\) and \(\left(\begin{array}{llll}1 & 2 & 3 & 4\end{array}\right)\). (Thus, \(S_{4}\) can be generated by a transposition and a 4 -cycle, but not every choice of transposition and 4 -cycle gives a generating set.)

(i) Prove that every group \(G\) of order 4 is isomorphic to either \(\mathbb{Z}_{4}\) or the 4-group \(\mathbf{V}\). (ii) If \(G\) is a group with \(|G| \leq 5\), then \(G\) is abelian.

Prove that a group \(G\) of even order has an odd number of elements of order 2 (in particular, it has at least one such element). (Hint. If \(a \in G\) does not have order 2, then \(a \neq a^{-1} \cdot\) )

Prove that every subgroup of a cyclic group is cyclic. (Hint. Use the division algorithm.)

Prove that two cyclic groups are isomorphic if and only if they have the same order.

If \(S\) and \(T\) are (not necessarily distinct) subgroups of \(G\), then an \((S-T)\) -double coset is a subset of \(G\) of the form \(S g T\), where \(g \in G\). Prove that the family of all \((S-T)\) -double cosets partitions \(G\). (Hint. Define an equivalence relation on \(G\) by \(a=b\) if \(b=\) sat for some \(s \in S\) and \(t \in T\).)

(Dedelkind Law). Let \(H, K\), and \(L\) be subgroups of \(G\) with \(H \leq L\). Then \(H K \cap L=H(K \cap L)\) (we do not assume that either \(H K\) or \(H(K \cap L)\) is a subgroup).