Answer to Question #44356 in Abstract Algebra for Jasvinder Singh

Question

5) a) Let H = h(1 2)i and K = h(1 2 3)i be subroups of S3. Check that S3 = HK. Is S3 the internal
direct product of H and K? Justify your answer.
b) Let s = 1 2 3 4 5 6 7
2 4 5 6 7 3 1and t = 1 2 3 4 5 6 7
3 2 4 1 6 5 7be elements of S7.
i) Write both s and t as product of disjoint cycles and as a product of transpositions,
ii) Find the signatures of s and t.
iii) Compute ts