$\mathbb{Z}_k \times \mathbb{Z}_l$ and $A_n$ Cayley's theorem says that $G = \mathbb{Z}_k \times \mathbb{Z}_l$ is isomorphic to some group in $S_n, n = kl$. Using programming I found, that $G \simeq P \leq A_n$ iff $k$ and $l$ have same parity, but how can I prove it mathematically?
 A: The image of the embedding $ \phi: C_k \times C_l \to S_{kl} $ defined by Cayley's theorem is generated by $ \phi(g_1, e) $ and $ \phi(e, g_2) $ (where $ g_1, g_2 $ are generators of $ C_k, C_l $ resp.) , so the image of this lies in $ A_{kl} $ iff both $ (g_1, e) $ and $ (e, g_2) $ as even permutations (by multiplication) on the group $ C_k \times C_l $. Now, note that $ (g_1, e) $ acts on $ C_k \times C_l $ as the product of $ l $ disjoint $ k $-cycles, and $ (e, g_2) $ acts as the product of $ k $ disjoint $ l $-cycles. The sign of these permutations are therefore $ (-1)^{(k+1) l} $ and $ (-1)^{k (l+1)} $. They are even permutations iff both of these signs are $ 1 $, so we need both $ (k+1)l $ and $ k(l+1) $ to be even, and subtracting yields that this happens iff $ k - l $ is even, i.e if $ k, l $ have the same parity.
Note that it is not true that $ A_{kl} $ cannot have a subgroup isomorphic to $ C_k \times C_l $ if $ k, l $ are of different parities - for example, the subgroup of $ A_{10} $ generated by $ (12)(34) $ and $ (56789) $ is isomorphic to $ C_2 \times C_5 $. This answer only says that the image of $ C_k \times C_l $ under the specific embedding into $ S_{kl} $ given by Cayley's theorem lies in $ A_{kl} $ if and only if $ k, l $ have the same parity.
