ROUP
ACTIONS
IDEA YOU CAN STUDY GROUPS BY MAKING THEM ACT
-
ON SPACES/SYSTEM
& VICEVERSA
,
YOU CAN BETTER UNDERSTAND
SPACES BY LOOKING AT GROUPS ACTING ON THEM
DEF
- -
An ACTION OF A GROUP G ON A SET X
is GxX >
- X
S T
(g x)kt
. -
x
g
.
,
(i) e . x = x Ve
(i) 91
.
(92 x)
:
=
(9192) ·
EXAMPLE (1)
-
G =
(e) Ex X - X TRIVIAL ACTION
(e x)
, +xx
(2)D u acts on the vertices of a
n-regular
polygen
wi
as follows
Us
- 3 Vg(i)
0
o =
.
LS -
,
v6 ,
3 3
rotation
by Et 0 02
.
= =
,
(3) GLm((R) acts on RRM
GLn(R) xR" >
- R2 usual multiplication
(A ,
8) 1
> Ao = Ar metrix vector
multiplication
(4) GRASSMANNIAN O-DIMENSIONAL
OF PLANES IN #M
IS THE SET
<H cRY/H
Gr(b n)
hearSubspese
=
GL(R) acts naturally on Gr(d n),
Gln((R) x Gr(d m) ,
>
- Gr(bin)
(A ,
H) += ACH) <I
Concretely H =
<21 , ...,
hdY
= AH = < Abs Ahdd
a
,
...,
PICTURE ON Gr(1 2) ,
A rotation by
- Ht
2-roots of unity (x21)
-
.
(5)
M2 <I 1) acts an
=
Mix 1
>
-
T
(5 n) ,
↳ En
I
5
-
- & ·
·
. . . · ⑳
:
·
-
W
. . .
I
- Z -
-
-
6) s =
32 e 4/11 1) =
e,
We have a lot of different actions of 1 on
St
.
Let DER .
So
Ex S1 gl
fo : -
2πimO
(n , z) +> ez
Nation We use G &X as a shorthand
for G action on X .
Ref G2X
THE IT of xeX is G .
x =
(g x/geG]
.
EXAMPLES
-
(3) GL
n (IR) 1/RM. We compute orbits
GLm(R) :
8 =
48)
Gt
.
(R) . =
(RM 1304 if &
() & &
M2
(2mhol
,
-
m M O
M2
.
m =
M = 0
() XS* S2 We DISTINGUISH 2 CASES
So
: >
-
.
(i) OeR so O = ab natural
numbers coprime
The orbit
Io := ei 4tie x
,
....
e zile al is finite
O=
Picture · orbit of 1
c · orbit of (,
(ii) O 0 ,
So IRRATIONAL
THEN THE ORBIT OF A POINT IS DENSE !
oppo
Pure
-e
orbit of 1
EXERCISE COMPUTE ORBITS IN EXAMPLES IN (1) , (2) (4)
,
.
We now introduce some natural motions for actions
group
RF G1 X .
We the ACTION IS RANSITIVE if Gox X
say
=
↳ E if x for Some =D id
g
x
g
. = =
EXAMPLES (2) ACTIONIS TRANSITIVE AND FREE
(3) NOT TRANSHIVE
,
NOT FREE
(4) TRANSITIVE NOT FREE
,
(5) NOT TRANSITIVE ,
NOT FREE
(6) NOT TRANSITIVE ,
FREE
ef Let X .
The STABILIZER OF is
Stab(x) =
(g G/g = .
x =
x)
Rem
- -
Action is free <> Stab (x) =
<e) for all x
EXERCISE COMPUTE STABILIZERS IN EXAMPLES
- -
(1) -
(6) .
PropAG 1X* 4 G : >
-
Perm (X) =
<f X-X) bijective)
:
g + (x 9 x) .
is a
group homomorphesm
If We to
hove
verify
(i) P (e) =
identity function
ok
by group actions exious
(ii) 4
(gh) = 4 (g) 4(2). Indeed
P(gh)(x) (gh) x (h x)
g
·
=
-
= .
L group
=
action
P(g)4(h)(x)
exious
EXERCISE G & X free
-
G
<D 4 :
- Perm (*)
injective
ennnnnnnnnnnnn
Note A
group COMES ALWAYS WITH INATURAL
ACTION ON ITSELF .
GIG
E LEFT MULTIPLICATION
GxG >
- G
(g ,
2)t gh
(2) CONJUGATION
GxG >
-
G
(gih) t
ghg-
Ercise Left multiplication is free & transitive
Hint : same as cancellation low & inverse
REMARK CONJUGATION CAN BE NEITHER
-
TRANS .
NOR FREE
(convince yourself)
Theorem (Cayleyaction
4 : G - Perm (G)
g
+ (4(g) : 2 +
gh)
is an
injective group homomorphesie
Pf home follows from Prop A
--
group
injectivity because action is free .
REM THIS SAYS THAT EVERY FINITE GROUP
is
-
A SUBGROUP OF PERMUTATIONS
