Suppose G is a finite group of order mn, where gcd(m,n)=1 and H is a normal subgroup of order m. Prove that H is the only subgroup of order m in G. Solution G IS FINITE GROUP ITS ORDER = M*N GCD[M,N] = 1 H IS A NORMAL SUBGROUP OF ORDER M TST H IS THE ONLY SUB GROUP OF ORDER M IN G . SINCE H IS A NORMAL SUB GROUP OF G , FOR EVERY X AN ELEMENT OF G AND FOR EVEY h AN ELEMENT OF H X*h*X INVERSE IS AN ELEMENT OF H PROOF BY CONTRADICTION LET H = [H1,H2,.