ABSTRACT ALGEBRA

QUESTION 2 (a) Show that the function h: R :- R defined by hlx} = 31: is an isomorphism of additive groups (b) Let G be a group with N, K as subgroup of G. Define NK ={nk\ E N,k E K} (i) If N is a normal subgroup of G, then prove that NK is subgroup of G {ii} If both N,K are normal subgroups of G, then prove that NK is a normal subgroup of G {c} Consider the additive group G = 22 x Z4 and [i] List all the elements of G [ii) If H = is cyclic subgroup, list the elements of H [iii] Using the question {ii} above find all the cosets of GIH [iv] Construct the addition table of the cosets OF GfH