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1. Write a complete Cayley Table for D6, the dihedral group of order 6. 2. Prove that if G is a group with property that the square of every element is the identity, then G is abelian. 3. Construct the Cayley table for the group generated by g and h, where g and h satisfy the relations g 3 = h 2 = e and gh = hg2 . 4. Let H and K be subgroups of a group G such that gcd(|H|, |K|) = 1. Apply Lagrange’s theorem to show that |H ∩ K| = 1. 5. Consider the group Z12 and the subgroup H =< [4] >= {[0], [4], [8]}. Are the following pairs of elements related under ∼H? Justify your answer. (a) [3], [11], (b) [3], [7], (c) [5], [11], (d) [6], [9], (e) find all left cosets of H in G. Are they different from the right cosets?
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