Let G a group such that x7 e ForAll x elementof G show that

Let G a group such that x^7 = e ForAll x elementof G show that G is abelian and |G| is finite and order of G is a power of 2 |G| = 2^k?

1. Given for all x in the group G : x² = e (ie) x*x =e ----(1)

but x * x^-1 =e ---(2)

from 1 and 2 it is clear that x= x^-1 ie every element has its own inverse

consider x , y in G then x*y also in G ( closure property )

x*y in G => (x*y) = (x*y)^-1 = y^-1*x^-1 = y *x (as every element in G has its own inverse)

x*y= y*x +. G is an abelian group

2 . Using Corollary of Lagrange\'s theorem

for any element \'x \' in G x^O(G) =e -----(I) ,

given x²=e and hence in general x²ⁿ=e ------(II)

comparing I and II we get O(G) =2n