Abstract Algebra

Abstract Algebra

1. Let G be a group and let G1 and G2 be subgroups of G. Assume that neither G1 contains G2 nor G2 contains G1. Prove that G1 UG2 is not a subgroup of G.

2. Let G be the set of all 2X 2 matrices of the form

[¦(a&0@0&b)],

wherea ? 0 and b ? 0, endowed with the matrix multiplication. Prove that it is a group. Find all self-inverse elements of this group. Justify.

3. Let G be a group of three elements G = {c, a, b}(all three elements are distinct). Prove that ab = ba= eand a^2= b.

4. Let abe a permutation of n elements and assume that a^3= e, where eis the identity permutation. Prove that ais an even permutation.
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