Chapter 6, exercise M.7: Let G be a finite group operating on a finite set S. For each element g of G, let Sº denote the subset of elements of S fixed by g: S⁹ = {s € Slgs s}, and let G, be the stabilizer of s. = (a) We may imagine a true-false table for the assertion that gs = s, say with rows indexed by elements of G and columns indexed by elements of S. Construct such a table for the action of the dihedral group D3 on the vertices of a triangle. (b) Prove the formula Σses |Gs| = Σ9€G |S⁹|. (c) Prove Burnside's Formula: G (number of orbits) = ΣgeG|S⁹|. .
Chapter 6, exercise M.7: Let G be a finite group operating on a finite set S. For each element g of G, let Sº denote the subset of elements of S fixed by g: S⁹ = {s € Slgs s}, and let G, be the stabilizer of s. = (a) We may imagine a true-false table for the assertion that gs = s, say with rows indexed by elements of G and columns indexed by elements of S. Construct such a table for the action of the dihedral group D3 on the vertices of a triangle. (b) Prove the formula Σses |Gs| = Σ9€G |S⁹|. (c) Prove Burnside's Formula: G (number of orbits) = ΣgeG|S⁹|. .
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:(4) Chapter 6, exercise M.7: Let G be a finite group operating on a finite set S. For
{s €
each element g of G, let Sº denote the subset of elements of S fixed by g: S⁹
S|gs = s}, and let G, be the stabilizer of s.
=
(a) We may imagine a true-false table for the assertion that gs :
=
s, say with rows
indexed by elements of G and columns indexed by elements of S. Construct
such a table for the action of the dihedral group D3 on the vertices of a triangle.
(b) Prove the formula Σses |Gs| = ΣgeG|S⁹|.
(c) Prove Burnside's Formula: |G|· (number of orbits) = Σ9EG |S⁹|.
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