...
H of index 2 ? — No (
H is normal in
G).
H of index ≥ 3 ? — Fix two cosets
Ha,
Hb of
H, distinct from each other and from
H ,
and define τ ∈ |
G|
! — an involution —
by giving τ(
x), for
x ∈
G, as
| |
|
{ |
xa−1b
(= h0b ∈ Hb) , |
if x = h0a ∈ Ha ; |
| τ(x) | = | x ,
|
if x ∉ Ha ∪ Hb ; |
| | |
xb−1a
(= h0a ∈ Ha) , |
if x = h0b ∈ Hb . |
Write κ
τ:
|
G|
! → |
G|
! for conjugation by involution τ —
κ
τ(σ) = τ·σ·τ . Compare left-regular
representation ρ: G → |G|! ({ρ(
g)}(
x) =
gx) with the composition κ
τ·ρ: G → |G|! → |G|! :
For
h ∈
H and
x ∈
G,
