(i) Let G=Z12(sub12 don't know how to put it), the group of integers modulo 12. Prove that H= {0, 6} AND K= {0, 4, 8} are subgroups of G. Calculate the subset H+K formed by adding together all possible pairs of elements from H and K, i.e.
H+K= {h+kh is a subgroup of H, k is a subgroup of K}
Prove that this is also a subgroup of G.

(ii) If H and K are subgroups of an abelian group G, prove that the set HK={hkh is subgroup of H, k is subgroup of K} is always a subgroup of G.
Show, by considering the group G=S, or otherwise, that the corresponding result is false when G is not abelian.

(b)

(i) in each of the group Z12(sub 12 should be placed at the corner below 12), and S3(3 is placing at the corner, as sub 3), list the elements of order 1 or two. In each case, does this set of elements from a subgroup?

(ii) Prove that in any abelian group G, the set {g/g²=e} is a subgroup of G.
Does this result remain true if G is not abelian? Justify your answer.

Solution Preview

Please see the attached file for the complete solution.
Thanks for using BrainMass.

Problem #3
(a) Proof:
(1) In the group , we consider the element . Since in , then the order of is 2. Thus is a cyclic subgroup of . Similarly, the order of is in . So is a ...

Solution Summary

Abelian Groups and Subgroups are investigated. The solution is detailed and well presented.

... H] = 9 then H must be a normal subgroup of G ... Homomorphism, Subgroups, Abelian Groups and Group Order are investigated. ... Math 3033 Group Theory Holiday Assignment ...

The Group Theory Concept: The Abelian Group. ... A group of order ,where is a prime number, is abelian. Solution:- Let be a group of order , that is . Then. ...

Group theory : explained. ... 8. You did not give the definition of the groups Z and V ... solution is comprised of a detailed explanation to solve group theories problem ...

Group Theory 1. i. State the axioms for an equivalence relation ii ... d,e) is either (or both) of the groups of order ... Let x and y be elements of a group, G. Prove ...

... The direct product of solvable groups is solvable ... It gives 5 interesting facts about group theory. ... 1. Every group of order n is isomorphic to a subgroup of the ...

... For the two homomorphisms in (c), prove that the two quotient groups G/N ... Many people get lost in group theory. ... OR x*ker h = {xj| j ε ker h} If the group G1 is ...

Group and Ring Theory Problems. Group problems. ... Some concepts of groups and rings are given. ... The group of automorphisms of a finite cyclic group is studied. ...

... this problem, we focused on the abelian group structure of Zn ... solution explains the answers for three problems in group theory. ... a fixed element in a group is an ...

Category Theory - Morphisms, Uniqueness & Equivalence. ... direct sum is not a coproduct in the category of all groups. ... the direct sum S3 ⊕ Z/2, a group of order ...

... we solve several problems in abstract algebra, mainly in group theory. ... ii) Prove that if G is an abelian group, then the ... of ﬁnite orders form a subgroup of G ...