Assignment:
Q1. Let H be a subgroup of a group G such that g ? ¹ hg elements in H for all h elements in H. Show every left coset gH is the same as the right coset Hg.
Q2. Prove that if G is an abelian group, written multiplicatively, with identity element e, then all elements, x, of G satisfying the equation x²=e form a sub group H of G
Q3. Show that if a elements in G where G is a finite group with the identity, e, then there exist n elements in Z+ such that a n =e
Q4. Prove the generalisation of the first part of this question: consider the set H of all solutions, x, of the equation x n =e for fixed integer n ≥1 in an abelian group, G with identity , e.
Q5. If is a binary operation on a set, S an element, x elements in S is an idempotent for . if x . x= x. prove that a group has exactly one idempotent element.
Q6. Define the term 'normal subgroup'. Give an example of a group, G, and a normal subgroup, H, of G
Q7. Prove that every group, G, with identity, e, such that x . x=e for all x G is abelian .
Q8. Draw the cayley tables for the Z and V. for each group, list the pairs of inverses
Q9. Determine whether the following are hohmorphisms. Let:
i. ? : Z → R under addition be given by ? (n) =n.
ii let G be any group and let : G→ G be given by ? (g)=g ? ¹
for g elements in G
Q10. Show that if G is nonablelian, the quot6ient group G/Z(G) is not cyclic ( i know that some how i have to show the equivalent contrapositive, ie that if G/Z(G) is cyclic then g is ablelian and hence Z(G)=G)
Q11. Show that the intersection of normal subgroups of a group G is again a normal subgroup of G
Provide complete and step by step solution for the question and show calculations and use formulas.