1. Let G=GL(2,Q), Q meaning rational numbers, and let A =matrix 0 -1
and B =matrix 0 1
Show that A^4 = I = B^6, but that (AB)^n does not equal I for all n >0. Conclude that AB can have infinite order even though both factors A and B have finite order( this cannot happen in a finite group)
Define W=[(1 2)(3 4)], the cyclic subgroup of symmetric group with n=4 (S sub 4) generated by (1 2)(3 4). show that W is a normal subgroup of V but that W is not a normal subgroup of S sub 4. Conclude that normality is not transitive: K is a normal subgroup of H and H is a normal subgroup of G need not imply K is a subgroup of G.
V= four group
I really dont know how to do these, I need to learn them in written proof form and I struggle.
This provides an example of a proof regarding a matrix with infinite order, and a proof regarding normal subgroups.