Prove that if the finite group G has an element of even orde

Prove that if the finite group G has an element of even order then G itself has even order Prove that the order of U(n) is even if n > 3.

Solution

a) Let |G|=2n|G|=2n. Since G is finite, there exists, aGaG such that a^p=e, and by Lagrange\'s Theorem, p divides 2n

. By Euclid\'s lemma, since p does not divide 2, p divides n. Let n=pkn=pk. Hence, (a^n)^2=(a^pk)^2=((a^p)^k^)2=(e^k)2=e^(a^n)^2=(a^pk)^2=((a^p)k)^2=(e^k)^2=e. Therefore, anan is an element that satisfy the condition

b)

Proof: n-1 has order 2 in U(N),so 2 divides the order of U(N).

(n-1)^2=(n-1)(n-1)=n^2-2n+1= 1 mod n

So the order of U(N) is even