A_n refers to the subset of the symmetric group S_n consisting of even permutations. (A permutation is even if it is the product of an even number of transpositions.)
\[ \text{Does }A_6\text{ have an element of order 6? Does }A_7\text{?} \]
Solution:
\( \displaylines{\text{The only permutations of order 6 in }S_6\text{ are 6-cycles or a 3-cycle with a 2-cycle.}\\ \text{Both of these are odd (the 6-cycles consist of 5 transpositions, the 3 and 2 cycles consist of 3 transpositions).}\\ A_6\text{ has no element of order 6}.\\ \\ A_7\text{ does contain elements of order 6, of the form 3-cycle with two 2-cycles. For instance,}\\ \left(12\right)\left(345\right)\left(67\right)} \)