91Ó°ÊÓ

Find the orbits and cycles of the following permutations: (a) \(\left(\begin{array}{lllllllll}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 2 & 3 & 4 & 5 & 1 & 6 & 7 & 9 & 8\end{array}\right)\). (b) \(\left(\begin{array}{llllll}1 & 2 & 3 & 4 & 5 & 6 \\ 6 & 5 & 4 & 3 & 1 & 2\end{array}\right)\).

If \(G\) has no nontrivial subgroups prove it must have prime order.

Prove that the homomorphic image of a solvable group is solvable.

In \(S_{3}\) show that there are four elements satisfying \(x^{2}=e\) and three elements satisfying \(y^{3}=e\).

If an abelian group has subgroups of orders \(n\) and \(m\), respectively, then show it has a subgroup whose order is the least common multiple of \(n\) and \(m\).

Prove that any group of order 15 is cyclic.

Prove that \(A_{5}\) has no normal subgroups \(N \neq(e), A_{5}\)