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

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SUMMARY

The discussion centers on the properties of cosets in nonabelian finite groups, specifically whether distinct elements and their inverses can occupy separate cosets of a proper subgroup. The participants explore the implications of the equation Sx = Sx^{-1} and its consequences for the element x. A key insight is that if Sx = Sx^{-1}, then multiplying both sides by x leads to the conclusion that Sx^2 = S, indicating that x^2 is an element of the subgroup S.

PREREQUISITES
  • Understanding of group theory, particularly nonabelian groups.
  • Familiarity with the concept of cosets and subgroups.
  • Knowledge of group operations and inverses.
  • Basic proficiency in mathematical proofs and logical reasoning.
NEXT STEPS
  • Study the properties of nonabelian groups in depth.
  • Research the concept of cosets and their applications in group theory.
  • Explore the implications of the subgroup structure in finite groups.
  • Learn about the significance of element orders in group theory.
USEFUL FOR

This discussion is beneficial for mathematicians, particularly those specializing in abstract algebra, as well as students studying group theory and its applications in various mathematical contexts.

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

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