Originally Posted by

**Deveno** here is a non-abelian group of order 8, the 4th dihedral group D4 = {1,r,r^2,r^3,s,rs,r^2s,r^3s} with r^4 = s^2 = 1, sr = r^3s.

we have the subgroup of order 4, {1,r,r^2,r^3} and clearly, for every element x in that subgroup, x^4 = 1.

(1^4 = 1, r^4 = 1, (r^2)^4 = (r^4)^2 = 1^2 = 1, (r^3)^4 = (r^4)^3 = 1^3 = 1).

so there are 4 elements right there. but s^4 = (s^2)^2 = 1^2 = 1, so s is a 5th element whose 4th power is 1.

when looking for counter-examples in groups, non-abelian groups of small order often work well.