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) ÎSX 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| Gx \\X | =(1)/( | G | ) åg ÎG | Xg | 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 thath:G -> G :g -> g'gg'-1,for a suitable g' ÎG. Show that this equivalence relation has the same classes as ».
|harald.fripertinger "at" uni-graz.at||http://www-ang.kfunigraz.ac.at/~fripert/||UNI-Graz||Institut für Mathematik||UNI-Bayreuth||Lehrstuhl II für Mathematik|