plz try to do these problem fast and clear!

Let G be a finite group, K a normal subgroup of G and P a Sylow subgroup of K. If N is the normalizer of P in G (i.c., N = (g e G \\ gpg ] = P} then show that G = K . N.Let P be a Sylow subgroup of a finite group G. Let N= (r E G | zPa-1= P). Let H be a subgroup of G, H 2 N. Prove: If y E G such that yHy ' = H, then y E H. (Indiana)Let G be a finite group. The probability of G to be commutative is defined by P(G) = H(a, b) E G x G | ab = ba}| IG X GI (a) If G is not commutative, prove that P(G) $ 5/8. (b) Give an example such that P(G) is exactly 5/8