Let G = GL2(R). {[ {[ а (a) Prove that T = | a, c, d e R, ad is a subgroup of G. d a (b) Prove that D = | a, d e R, ad + 0 is a subgroup of G. 0 d a (c) Prove that S | a, b, c, d E R, b = c} is not a subgroup of G. C d

Need help with this abstarct algebra question

Let H and K be subgroups of a finite group G. Show that |HK |HK= |HОКI where HK (hk hE H, k E K}. (HK is not assumed to be a subgroup of G) :

Show that the group of rotations in R³ of a 3-prism (that is, a prism with equilateral ends, as in the following figure) is isomorphic to D3.

Let G = GL2(R). Prove that each of the following subsets is a subgroup of G. for а (а) Н %— a, b e R, a + 0 0 a-1 (b) K = {A € G | det(A) E {±1}}. 1 1 A 1 0 1 1 (c) L = 1 0