solve 1 and 2

1. (a) Define what it means for a group H to be a cyclic subgroup of a given group G. You must define what it means for a group to be cyclic as part of your answer. (b) Give an example of a group G which is not cyclic but has a cyclic subgroup H of order >1. No proof required, but be sure to completely define the group and subgroup. You may use standard notation from the textbook, if needed. 2. (a) Give the definition of an isomorphism from a group G onto a group H. (b) Suppose :GH is an isomorphism from a group G onto a group H. Prove that if K is a subgroup of G, then (K)={(g):gK} is a subgroup of H. 3. (a) Define what it means for a permutation to be even or odd. (b) Determine, with proof, whether the two products of cycles (1,2,3)(1,2) and (1,2,3,4,5)(1,2,3)(4,5) are even or odd