Exercises

Exercises

E: Assume X to be a G-set and check carefully that g -> bar (g) is in fact a permutation representation, i.e. that bar (g) ÎS_X and that bar (g₁ ·g₂)= bar (g₁) ·bar (g₂).

E: Prove that ~_G is in fact an equivalence relation.

E: Let _GX be finite and transitive. Consider an arbitrary x ÎX and prove that

| G_x \\X | =(1)/( | G | ) å_{g ÎG} | X_g | ².

E: Check that the G-isomorphy simeq (and hence also the G-similarity ») is an equivalence relation on group actions.

E: Consider the following definition: We call actions _GX and _GY inner isomorphic if and only if there exists a pair ( h, q) such that _GX simeq _GY and where h is an inner automorphism , which means that

h:G -> G :g -> g'gg'^-1,

for a suitable g' ÎG. Show that this equivalence relation has the same classes as ».

Exercises