Consider $S_5$, the symmetric group of degree five. Does it have a subgroup isomorphic to $C_5 \times C_5$? Does it have elements of order 6? Does it have a subgroup isomorphic to $D_5$? What about a subgroup isomorphic to $D_6$?
Is there an actual method to 'working out' this question or am I just expected to look up the answer and write yes or no for each part? I looked it up here - http://groupprops.subwiki.org/wiki/Subgroup_structure_of_symmetric_group:S5 - and see that it does have an element of order 6, namely $(1,2,3)(4,5)$. In the subgroup section it doesn't mention anything about subgroups isomorphic to $C_5\times C_5$, $D_5$ or $D_6$ so I take it doesn't have subgroups isomorphic to those groups?
And again, just to clarify, is there a method to working this question out or am I correct just to look it up?