I was curious to know, say we have two even permutations taken out of A_4, say (12)(34) and (123), and we want to find the smallest subgroup of A_4 that contains both these permutations, then how would we go about it. This subgroup in this case will defenitely be A_4 itself, here is how i came to this conclusion. SInce that subgroup should contain both these permutations, then it also should contain the subgroups generated by those permutations, and also the elements that are derived when we multiply these by each other, i kept going this way, and i finally generated the whole A_4. BUt imagine if we were working with a group of higher order, since ordA_4 =12, then this would be a pain. SO my real question is this, is there any clever way of finding these subgroups that contain, like in this case, two other elements. If it were just for one, i know that the smallest subgroup would be the cyclic subgroup generated by that element, but what about this case??? Any input is greately appreciated.