Actions of groupsSimilar ActionsExercises

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) 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 that
h: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
last changed: January 19, 2005

Actions of groupsSimilar ActionsExercises