Exercise 5.3. Let G be a group and H, K be two subgroups of G. (1) Show that HK := {hk | h € H, k = K} is a disjoint union of some left cosets of K. (2) Let H, K be subgroups of G. Show that we have the following bijection (between coset spaces) K/(H^K) → HK/K defined by к(HK) → KK, Vк(HK) Є K/(H□K), where HK/K := {left cosets of K in HK}. (3) Suppose [G: H] and [G: K] are both finite. Prove or disprove that [G: HK] is finite.

17. Let Dn = {1, a, ...‚ añ−¹, b, ba, … 2 (a) Show that every subgroup K of (a) is normal in Dn. (b) If n is odd and K ◄ Dn, show that K = Dn or K C (a). ‚ban-¹} with o(a)= n, o(b) = 2, and aba = b.

Determine whether or not H is a subgroup of G.

Let G = GL(2, R) and let g = (a) Letz 8 J Write down all the elements of the right coset Kr. 3 (b) Let x = a C d in terms of a, b, c, d. Let K(g) be the cyclic subgroup generated by g. b] be an element of GL(2, R). Write down all the elements of the right coset Ka