# alternating group

• Dec 11th 2012, 10:56 AM
MarcoRo
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
• Dec 11th 2012, 06:22 PM
Deveno
Re: alternating group
we can answer this completely just by looking at the possible disjoint cycle types in A5:

5-cycles (which can be written as a product of 4 transpositions: (a b c d e) = (a e)(a d)(a c)(a b))
3-cycles (which can be written as a product of 2 transpositions: (a b c) = (a c)(a b))
2 disjoint 2-cycles 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 5-cycles, etc. there are in S5, since for any cycle type in A5, A5 contains every cycle of that type.
• Dec 11th 2012, 11:07 PM
MarcoRo
Re: alternating group
Thank you, Deveno!

Can this be used to deduce hat A_5 cannot have a nontrivial subgroup?

Marco