I have the following questions but I don't totally understand them, so im not sure about it all:
Let H be the subgroup of
generated by the permutations
(1) Find the order of
. What's the order of
?
(2) Find the order of H.
(3) Find the conjugacy classes of H.
(4) Show that
is a normal subgroup of H
(5) To which well-known group is the quotient H/Z isomorphic?
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For (1) I know the orders are all 4 I think.
For (2) Is the fastest way of doing this to actually go through and compute every new element?
Or is there an easier way of finding the order?
For (3) I think the conjugacy classes are the sets in which the permutions all have the same cycle structure?
For (4) Z is normal iff Z is a union of conjugacy classes?
For (5) I think this is isomorphic to the Klein Four Group. Find the orders of elements in H/Z ?
Thanks for any help.