Question

5.

(a) Let σ = (1 2 3 4 5 6) in S6. Show that G = {ε, σ, σ^2, σ^3, σ^4, σ^5} is a group using the operation of S6. Is G abelian? How many elements τ of G satisfy τ^2 = ε? τ^3 = ε? ε is the identity permutation.

(b) Show that (1 2) is not a product of 3-cycles. Must be written as a proof!

(c) If a^4 = 1 and ab = b(a^2) in a group, show that a = 1. Must be written as a proof!

(d) Show that a group G is abelian if and only if (gh)^2 = (g^2)(h^2) for all g and h in G. Must be written as a proof!

Answer 1