where denotes the projective general linear group of degree two, denotes the projective special linear group of degree two, and denotes the special linear group of degree two.
The claim is that all these isomorphic groups are isomorphic to alternating group:A5.
(S3) A5 is the unique simple non-abelian group of smallest order: is a simple non-abelian group of smallest order and any simple non-abelian group of the same order is isomorphic to . Used to force any other simple non-abelian group of that order to be isomorphic to it.
(S4) Projective special linear group is simple (non-abelian) with some exceptions for small sizes and degrees is simple.
(E2) Equivalence of definitions of size of projective space Used to calculate size of projective space of dimension one over field with four elements.
(E4) The only subgroup of order 60 in symmetric group:S5 is alternating group:A5 as A5 in S5. For more, see subgroup structure of symmetric group:S5. Determining the image of an injective group homomorphism to show isomorphism to the image.
The proof follows directly by combining (S1)-(S4) as follows: (S1) and (S2) establish that and both have order 60. (S3) and (S4) establish that both are simple non-abelian, but is also the only one up to isomorphism for order 60 by (S3). This forces them to be isomorphic.
3 For the field of size , has order and is the symmetric group of degree and has order . (E2), (E3) Steps (2), (3) Plug in and evaluate.
5 For the field of size , the homomorphism of Step (1) gives an injective group homomorphism from to whose image is the subgroup . In particular, is isomorphic to . Steps (1), (3), (4) By Steps (1) and (3), we get an injective homomorphism from to whose image has order the same as , which is 60. By Step (4), the only possibility for the image is the subgroup , completing the proof.
This page was last edited on 30 April 2012, at 23:04.