Why treat index 2 as special case?
Why mix left regular representation of
G with right cosets of
H?
If you need room for a good involutory permutation τ, just make room —
add another point ∞ to the space
G/
H of left cosets
gH of
H ,
forming
E =
G/
H ∪ {∞} ,
have τ ∈
E! interchange
H with ∞ but leave the rest of
G/
H alone.
Then do as before:
Write κ
τ:
E! →
E! for conjugation by the involution τ —
κ
τ(σ) = τ·σ·τ ;
compose left-regular
representation
ρ:
G → (
G/
H)
! ⊂
E!
({ρ(
g)}(
xH) =
gxH , {ρ(
g)}(∞) = ∞ )
with κ
τ:
E! →
E! ;
for
h ∈
H and
C ∈
G/
H ∪ {∞}
(
C =
xH
(
x ∈
G) or
C = ∞ ) ,
calculate {κ
τ·ρ(
h)}(
C) =
| |
{ |
τ({ρ(h)}(τ(xH))) = |
τ({ρ(h)}(xH)) |
= τ(hxH) = hxH |
= ρ(h)(C) , |
(C = xH ≠ H) ; |
| τ({ρ(h)}(τ(C))) = |
τ({ρ(h)}(τ(H)))
= τ({ρ(h)}(∞))
= τ(∞) = H = {ρ(h}(C) , |
(C = H) ; |
| |
|
(C = ∞) . |
