Let G = { σ1, ..., σn} be a finite group with the σi the regular representation of G. Denote C(G) as the Cayley table of G (as a matrix) and Pα the permutation matrix of the permutation α.
Then C(G) = σ1 Pτ1 + ... + σn Pτn for some { τ1, ..., τn } ∈ Sn. Define φ(σi) = τi in this way.
Let f : G → G ; f( x ) = x− 1. This function is obviously bijective, and denote μ(G) be the permutation defined by the action of f on G.
(To be proven) Then φ(σi) = μ(G)σi.
Corr. Let G = (Z2)m. Then φ(σi) = σi.
Pf. In (Z2)m, every element is a product of transpositions and thus self-inverse. Thus μ(G) = e.
Clearly, every φ(σi) is of the form (1 i)υi, for some υi. Define φ'(σi) = υi, ie., φ'(σi) = (1 i)μ(G)σi.
(To be proven) < φ'(σ1), ..., φ'(σn) > = Sn.
Then C(G) = σ1 Pτ1 + ... + σn Pτn for some { τ1, ..., τn } ∈ Sn. Define φ(σi) = τi in this way.
Let f : G → G ; f( x ) = x− 1. This function is obviously bijective, and denote μ(G) be the permutation defined by the action of f on G.
(To be proven) Then φ(σi) = μ(G)σi.
Corr. Let G = (Z2)m. Then φ(σi) = σi.
Pf. In (Z2)m, every element is a product of transpositions and thus self-inverse. Thus μ(G) = e.
Clearly, every φ(σi) is of the form (1 i)υi, for some υi. Define φ'(σi) = υi, ie., φ'(σi) = (1 i)μ(G)σi.
(To be proven) < φ'(σ1), ..., φ'(σn) > = Sn.