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  3. 5. (a) let ? = (1 2 3 4 5 6) in s6. show that g...

5. (a) Let ? = (1 2 3 4 5 6) in S6. Show that G...

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Advance Math

5.

(a) Let ? = (1 2 3 4 5 6) in S6. Show that G = {?, ?, ?^2, ?^3,?^4, ?^5} is a group using the operation of S6. Is G abelian? Howmany elements ? of G satisfy ?^2 = ?? ?^3 = ?? ? is the identitypermutation.

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

(c) If a^4 = 1 and ab = b(a^2) in a group, show that a = 1. Mustbe 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!

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