Let G be a finite group with no element of order 3, and let a E G. (a) Can 3 divide o(a)? Either prove that it cannot or give an example where it does. (b) Must there exist an element x E G with x = a? Either prove that there must or give an example where there is not. (c) Must there exist an element y E G with y² = a? Either prove that there must or give an example where there is not.

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Q: (a) Show that a group G is abelian, if (ab)² = a²b², for a, b € G[C.

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Q: 10. Let G = Z/36Z. (a) Find the order of every element of the group G. (b) Find all generators of G.

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Q: For matrix A = (4 6 3 5 ). If f(x) = x2 - 5, calculate f(A)?

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