Questions Involving Cosets

1. Let G = Q8 (quaternion group). Find the right cosets of H in G for:

a) H = <J>

b) H = <-I>

2. Find the right cosets of the subgroup H = <(1,1)> in Z2 x Z4

3. Let H be a subgroup of a group G and define H congruent on G by letting x congruent to y IFF x^-1y is in H.

a) Show that H congruent is an equivalence relation on G

b) Show that the equivalence classes under H congruent are the left cosets of H in G.

c) Show that for a,b in G, aH = bH iff a^-1b is in H.

4. Let G = Q8 (quaternion group). Find [G : H] for H = <-I>, H = <K>, and H = <-L>.

5. a) In G = (Z48, +), find [G : H] for H = <32>.

(+ is actually addition mod 48, if it was unclear). There are two more parts to this question but if someone could tell me how to do this one, I can follow with the others.