Purchase Solution

Group Theory: Abelian Groups and Subgroups

Not what you're looking for?

Ask Custom Question

3 (a)

(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.

Purchase this Solution

Solution Summary

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

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 ...

Purchase this Solution


Free BrainMass Quizzes
Graphs and Functions

This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.

Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

Solving quadratic inequalities

This quiz test you on how well you are familiar with solving quadratic inequalities.

Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts