Let G be a finite group and let x be an element of G Prove t

Let G be a finite group and let x be an element of G. Prove that x^n=e for some positive integer n.

Solution

It is give that G is finite Group,

Let G = {g1, g2, · · · , gp} for some positive integer p.

For some gi G,

Sequece would be gi , g^2 i , g^3 i , · · · .

Since G is finite and closed under binary operation, so there must be repetition in the sequence,

thereore g^ x i = g ^y i for some positive integers x and y with x > y.

But that means g^(x-y) i = e, where e is the identity element.

Again Let x y = ni

we have ni corresponding to every gi such that g^(ni) i = e

Let N = n1×n2×· · ·×np.

But then g^n i = e for all i

showing the existence of required positive integer n.

Proved