The only non-abelian simple group of order 60 is A_5

Hi, assuming that a simple non-abelian group of order 60 can be embedded in , I want to prove that cannot contain an odd permutation and is therefore a subgroup of (and then from that I can deduce that as it is order 60 it is ). How would I go about showing this?

Re: The only non-abelian simple group of order 60 is A_5

Quote:

Originally Posted by

**alsn** Hi, assuming that a simple non-abelian group

of order 60 can be embedded in

, I want to prove that

cannot contain an odd permutation and is therefore a subgroup of

(and then from that I can deduce that as it is order 60 it is

). How would I go about showing this?

Consider the sign map where we're thinking of . Since is simple this is either zero or an embedding, and since I'm pretty sure it's clear that the map is zero. But, this tells us precisely that every element of is even.

Re: The only non-abelian simple group of order 60 is A_5

Quote:

Originally Posted by

**Drexel28** Consider the sign map

where we're thinking of

. Since

is simple this is either zero or an embedding, and since

I'm pretty sure it's clear that the map is zero. But, this tells us precisely that every element of

is even.

ahhh okay. so since all the elements of G map to zero and none map to 1, does this mean that all combinations of elements of G are even? think i get it.

Re: The only non-abelian simple group of order 60 is A_5

Are you intending to make a distinction between "elements of G" and "combinations of elements of G"? The nature of a group is that combinations of elements are just elements themselves.

Re: The only non-abelian simple group of order 60 is A_5

Quote:

Originally Posted by

**alsn** ahhh okay. so since all the elements of G map to zero and none map to 1, does this mean that all combinations of elements of G are even? think i get it.

Like **Tinyboss** points out, this doesn't make sense per se. What I proved was that every element of 's embedding in has even sign and so the embedding lives inside .