Show that all simple groups of order 60 are isomorphic to A5.

[glacier302]

Prove that if G is a simple group of order 60, then G is isomorphic to A5.

So far, I have shown that there is a subgroup of G with index 5. I know that with this information I should be able to show that G is isomorphic to A5, but I'm not sure how...

 

[glacier302]

Prove that if G is a simple group of order 60, then G is isomorphic to A5.

So far, I have shown that G has a subgroup of index 5. If I call these 5 cosets K1, K2, K3, K4, and K5, then G acts by conjugation on these cosets, and there is a homomorphism from G to S5. I'm not sure how to proceed from there...

 

[vr88]

I think this is right.
As you said, there's a homomorphism from G to S5. The image of the G is a subgroup of S5, call it H. This subgroup can't be S5 itself, since S5 has 120 elements, and A5 has only 60. If |H| < 60 then the kernel of the homomorphism is more than just the identity, and is normal in G. But G is simple, so |H|=60, i.e. H=A5, so then G=A5.

 

[mathwonk]

i guess you should check that A5 is the only subgroup of order 60 in S5.

 

[glacier302]

Thank you, I think I figured it out : )