Can Elements and Their Inverses Occupy Distinct Cosets in Nonabelian Groups?

[nbruneel]

Hi,

This is not a homework question. I am a trying to prove a result for myself, and the question is can I always find, in a nonabelian finite group G, and some fixed proper subgroup S < G, two distinct elements, which we shall call x and y, outside of S, such that the cosets Sx = Sx^{-1}, and Sy = Sy^-1. That is, can we always find elements x, y outside of S such that x and its inverse x^{-1} both belong to some coset of S, while y and y^{-1} belong to a different, disjoint coset of S.

Nici.

 

[Office_Shredder]

If Sx=Sx^-1 multiply both cosets by x to get Sx²=S. What does this tell us about x²?