Let Dg be the Dihedral group of order 8. Prove that Aut(D8) = D8.

Let e € Sn be the identity. Assume that e = B1B2. Br, where B₁, B2,... Br are 2-cycles. Choose the correct option from below and explain why. (a) r is always odd. (b) r is always even. (c) r can be odd or even, it depends on the specific decomposition that we use.

Give Z(G), the center of G, for G = S3

If x and y are 3−cycles in Sn, prove that ⟨x, y⟩ is isomorphic to Z3, A4, A5 or Z3 × Z3.