Brian Mere's Digital Garden
Stabilizers are Subgroups
Stabilizers are Subgroups
G
s
≤
G
(see
Stabilizers
).
Proof
1
G
⋅
x
=
x
⇒
1
G
∈
G
s
(inverses) Suppose
g
∈
G
x
, so
g
⋅
x
=
x
. Then:
⇒
g
−
1
⋅
(
g
⋅
x
)
=
g
−
1
⋅
x
⇒
(
g
−
1
g
)
⋅
x
=
g
−
1
⋅
x
⇒
x
=
g
−
1
⋅
x
So then
g
−
1
∈
G
x
.
3. (products) Suppose
g
1
,
g
2
∈
G
x
. Then:
(
g
1
g
2
)
⋅
x
=
g
1
⋅
(
g
2
⋅
x
)
=
g
1
⋅
x
=
x
So
g
1
,
g
2
∈
G
x
.
☐