Proving Cosets of Subgroups in Nonabelian Finite Groups

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SUMMARY

The discussion centers on proving the existence of distinct elements x and y in a nonabelian finite group G, outside a fixed proper subgroup S, such that the cosets Sx = Sx^{-1} and Sy = Sy^{-1}. The example provided involves the group Z/6 with the subgroup S = {0, 2, 4}, illustrating that certain cosets are not disjoint. The conclusion drawn is that while specific cases may show disjoint cosets, the general proof remains to be established for all nonabelian finite groups.

PREREQUISITES
  • Understanding of group theory concepts, particularly nonabelian groups.
  • Familiarity with cosets and subgroup properties.
  • Knowledge of finite group structures and examples, such as Z/6.
  • Basic proficiency in mathematical proofs and logical reasoning.
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  • Study the properties of nonabelian finite groups and their subgroups.
  • Research the concept of cosets and their applications in group theory.
  • Examine specific examples of nonabelian groups to identify patterns in coset behavior.
  • Explore advanced topics in group theory, such as Sylow theorems and their implications for subgroup structures.
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Mathematicians, students of abstract algebra, and anyone interested in advanced group theory concepts, particularly those exploring the properties of nonabelian finite groups.

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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Consider Z/6 with S= {0,2,4}. S1(=S5) & S3 are not disjoint.
 

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