Answer to Question #72268 in Abstract Algebra for Lastone

Question

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?

Answer 1

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