MAT3702 Assignment 1 (DETAILED ANSWERS) 2026 - DISTINCTION GUARANTEED

MAT3702 Assignment 1 (DETAILED ANSWERS) 2026 - DISTINCTION GUARANTEED - DISTINCTION GUARANTEED - DISTINCTION GUARANTEED Answers, guidelines, workings and references.. 1. Let A,B,C be sets and show that A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C). 2. Let G be a finite group with x, y ∈ G and prove that xy and yx have the same order. 3. Let G be a group such that every nonidentity element in G has order 2 and prove that G is abelian. 4. Show that ( ZZ, +) is a subgroup of (Q| , +). 5. Show that A3 is a cyclic group of order 3. 6. Let G be an abelian group with a ∈ G and N a subgroup of G. Then show for every n ∈ N that an = na. 7. A group G is called metabelian if G has an abelian normal subgroup N such that G/N is abelian and hence show that S3 is metabelian. 8. Let G be a group with K a normal subgroup of G of order 2 and show that K ⊆ Z(K). 9. Let |R∗ be the multiplicative group of nonzero real numbers and check as to whether the function f : |R∗× |R∗ −→ |R∗ defined by f (x, y) = y (under componentwise multiplication) is a homomorphism or not and if it is a homomorphism, then find its kernel. 10. Let f : G −→ H be a homomorphism of finite groups. If Im(f ) is the image of f , then show that |Im(f )| divides both |G| and |H|.