Question

let g be a group. Call an isomorphism from G to itself a self similarity of G.

a) show that for any g ∈ G, the map c_g : G-> G defined by c_g(x)=g^(-1)xg is a self similarity group

b) If G is cyclic with generator a, and sigma is a self similarity of G, prove that sigma(a) is a generator of G

c) How many self-similarities does Z have? How many self similarities does Z_n have? How many self-similarities does Z₂ x Z₂ have?

d) Recall that D₄ , the group of symmetries of a square, is generated by two elements r and f satisfying relations r^4=1, f^2=1 and fr=(r^3)f. In class we saw that D₄ has two subgroups isomorphic to the Klein-4 group, namely X={1,f,r^2,r^2f} and Y={1,rf,r^2,r^3f}. Find a self-similarity of D₄ that exchanges X and Y

Please help me...

Answer 1