Exercises |

E:AssumeXto be aG-set and check carefully thatg -> bar (g)is in fact a permutation representation, i.e. thatbar (g) ÎSand that_{X}bar (g._{1}·g_{2})= bar (g_{1}) ·bar (g_{2})

E:Prove that~is in fact an equivalence relation._{G}

E:Letbe finite and transitive. Consider an arbitrary_{G}Xx ÎXand prove that| G_{x}\\X | =(1)/(| G |)å_{g ÎG}| X_{g}|^{2}.

E:Check that theG-isomorphysimeq(and hence also theG-similarity») is an equivalence relation on group actions.

E:Consider the following definition: We call actionsand_{G}X_{G}Yinner isomorphicif and only if there exists a pair( h, q)such thatand where_{G}X simeq_{G}Yhis aninnerautomorphism , which means thatfor a suitableh:G -> G :g -> g'gg'^{-1},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 |

Exercises |