Question

3 (Due 8/7) Prove that every element of a group has a unique inverse. (Due 8/7) Let (G, *) be a group and let a be an element of G with inverse d'. Prove that the function f(x) = a*r*d' is a permutation of G.

Answer 1

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Jo show that f is one-one: Let MY FG such that finy z fly) 3 Axtra") = ay are a q** = axy I By Right cancellation in 4) Ray I By left cancellation in as - f in One-one Do show that f is onto : let neg, I a't naq E G such that fla'x x*q) = 919'*x*qla's Caxa!)xXxlaqli = exkee =n. a f is onto. 4 clearly flus = 4*x*q' is closed under composition ad QE G, g'Eg als x F G =j fruje a. Therufen flas= a*K*q' ſ Permutation.