Abstract algebra

(2) Left versus right. For a given subgroup of a. given group, sometimes the collection of left cosets is identical to the collection of right cosets, and sometimes this is not the case. The problem illustrate: this. (a) List all the right cosets of H = ((12)) in $3. Show there is at least one right coset that is not equal to any of the left case-ts. \\L 0...- .. 4- _.' -4- in"- \"A-\" g 4...\"; (b) List all the left cosets of the subgroup K := ((123)) in 33 and also list all of the right cosets of K in 33. Show that these two lists are identical. (3) Let H and K be subgroups of a group G, and recall that you proved in the past that H n K is also a subgroup. Prove 9(H K) = 9H 0 9K for all g E G