E xamples o of Groups The integers mod p under multiplication G=Z'n=the non-zero integers modulo p=(1.p-1) the group operator is "*>, modular multiplication the integers modulo p are closed under multiplication this is so because if GCD(x, p)=l and gCd(y p)=1 then gCd(xy,p)=1 the identity is 1 the inverse of x is from euclids algorithm ux+ vp=1=GCD(x,p) SOX=u alsoxl=u=xp-2 multiplication is associative multiplication is commutative(so the group is abelian)Examples of Groups The integers mod p under multiplication G = Z* p = the non-zero integers modulo p = {1 … p-1} the group operator is “*”, modular multiplication • the integers modulo p are closed under multiplication: this is so because if GCD(x, p) =1 and GCD(y,p) = 1 then GCD(xy,p) = 1 • the identity is 1 • the inverse of x is from Euclid’s algorithm: ux + vp = 1 = GCD(x,p) so x-1 = u also x-1 = u = xp-2 • multiplication is associative • multiplication is commutative (so the group is abelian)