Let X = (v,w,x,y) and let G = Sym(X) Part A: Find all cyclic subgroups of G of o
ID: 3108277 • Letter: L
Question
Let X = (v,w,x,y) and let G = Sym(X)
Part A: Find all cyclic subgroups of G of order 4.
Explanation / Answer
I posted a comment for clarification, but I think this might be the heart of what the question is getting at. First and foremost, we know that it must be a 4-cycle, because any of the 2-cycles will have an order of 2, and we need an order 4 cyclic subgroup, same goes for 3-cycles. To represent a symmetric group on 4 elements, we can use numbers, such as (1234) representing 1->2, 2->3, 3->4, 4->1. Notice that this could be used as a basis for a cyclic subgroup of G if it is indeed the set of all permutations of X. We can do that by having: {(1234), (13)(24),(1432), (identity)} But as far as finding all of them, I think you're stuck just trying all possible combinations. Fortunately there aren't many, because remember, in this notation, (1234) is the same as (2341) and (3412), and (4123), so there are only 6 to try. (1234) (1243) (1324) (1342) (1423) (1432) (1234) works, as we found above (1243) times itself is (14)(23), but (14)(23)(14)(23) = (id), so we're okay there, and (1243)(14)(23) = (1342). With (1243)(1342) = (id), we have a cyclic subgroup again, it is: (1243),(14)(23), (1342), and (id) So I think at this point it should be fairly obvious that the (1342) showing up isn't a coincidence, and in fact, you'll notice that (1342) can also be a generator for this group. The same is true for our first one, the {(1234), (13)(24),(1432), (identity)} group having 1432 in it as well. I bet you can guess that last discovery we'll make about (1324) and (1423). So I'll leave you with the first two completely done and that very strong suggestions to try for that last one. Assuming I'm interpreting the question correctly, these will be the only three cyclic subgroups of the symmetric group of degree 4 (because of what I discussed earlier with the order 4 making it necessary to have a 4-cycle). Hope that helps!
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