Question

Exercise 2. Let he a group anith nentral element e. We denote the gronp lau on G simply by (91,92)gig2. Let X be a set. An action ofG on X is a a map that satisfies the following tuo conditions: c. Let G be a finite group. For each E X, consider the map (aje- fer all elements r X (b) 9-(92-2) for all 91,92 G and all r E X Show that is surjective and that, for all y E G , the cardinal of G.. Deduce that ) is equal to Given an elemeat x X, its orbit is the subset and its stabiliser is the subset of X and the cardinal of the tixed pe congruert modulo p, that is: XX° modulo p. The set of fized points XG is the subset Rind fr s! eTply is.grer ge's tha mem 'n the inhyr mp ("z fCand the identityG-IC, .G. Tİ frym yucation c. tc deduce tha! G 떠 ia α pouer ofp. Then distingwish the cesea tehcrc IG 21-1 axd where G r 2pend condude asing the owla frowt yuestioa b. a. Prove that, for any element E X, the stabiliser G is a subgroup of G and that the obit G z consists of a single element if a only if EX b. Prove that the binary relation cn X defined by ~y if and only if there exists ge G such that -g.a is an equivalence relation and that the equivalence class of an element r X is precisely its orbit G G. Deduce that, if X is a finite set and S denotes a set of representatives of the equivalence classes, we have

Answer 1

let 3..1, ε9. then "esh that 3-3: ε But this is clear since (b) Clearly since some Suppose Such that hus is an equiualence relat ions equivalence dlasses partition the since Z E x EX et e then take j ' e then Now since兀 surjective thus if yet en 兀 Surjecfiu

but l II-p since Ganges e orem n -k ラ ZEx t EX