
alternating group
Hello!
Can somebody help me with the following problem? Let A_5 be the alternating group of order 5.
a) Which orders can a given element of A_5 possibly have?
b) Let n be a possible order as in a). How many elements of order n does A_5 have?
This seems to work with the Sylow theorems, but how can one proceed?
Marco

Re: alternating group
we can answer this completely just by looking at the possible disjoint cycle types in A_{5}:
5cycles (which can be written as a product of 4 transpositions: (a b c d e) = (a e)(a d)(a c)(a b))
3cycles (which can be written as a product of 2 transpositions: (a b c) = (a c)(a b))
2 disjoint 2cycles such as (a b)(c d)
the identity (which can be written as a product of 2 transpositions: e = (a b)(a b)).
which have orders 5,3,2 and 1, respectively.
the second part of your question can then be answered by asking how many 5cycles, etc. there are in S_{5}, since for any cycle type in A_{5}, A_{5} contains every cycle of that type.

Re: alternating group
Thank you, Deveno!
Can this be used to deduce hat A_5 cannot have a nontrivial subgroup?
Marco