Groups
Stabilizers
Algebra
Interactive
Algebra 2
Groups
Stabilizers
A.M. Cohen, H. Cuypers, H. Sterk
A.M. Cohen, H. Cuypers, H. Sterk
Algebra 2
September 25, 2006
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Groups
Stabilizers
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The behaviour of a permutation representation
f : G → Sym(X ) can be recorded inside G . The
first step is to relate a point x of X to a particular
subgroup of G .
A.M. Cohen, H. Cuypers, H. Sterk
Algebra 2
September 25, 2006
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Groups
Stabilizers
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Definition
If x∈X , then the stabilizer of x in G is the subgroup Gx of G given by
Gx ={g ∈G | g (x)=x }.
If g (x)=x, then g is said to fix or to stabilizex.
A.M. Cohen, H. Cuypers, H. Sterk
Algebra 2
September 25, 2006
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Example
G =Sym5 acting on {1, 2, 3, 4, 5}
The stabilizer of 3 consists of all permutations g with g (3)=3. These are all permutations of {1, 2, 4, 5}. Hence, the stabilizer
is Sym({1, 2, 4, 5}), which is isomorphic with
Sym4 .
A.M. Cohen, H. Cuypers, H. Sterk
Algebra 2
September 25, 2006
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Example
G =Sym5 acting on the set X of subsets of
{1, 2, 3, 4, 5} of size 2
The stabilizer of the set {4, 5} consists of
all elements g of Sym5 with g (4)∈{4, 5} and
g (5)∈{4, 5}. In the disjoint cycle decomposition of such an element g , we find either the
cycle (4, 5), or no cycle at all in which 4 or
5 occurs. Thus, such an element g is either
of the form h or h·(4, 5), for some h∈Sym3 .
Hence, the stabilizer of {4, 5} is the subgroup
Sym3 × Sym({4, 5}). More precisely, the stabilizer is the image of the natural morphism Sym3 ×
Sym({4, 5})→Sym5 , [g , h] 7→ g ·h. Thus, the
stabilizer has order 6·2=12.
A.M. Cohen, H. Cuypers, H. Sterk
Algebra 2
September 25, 2006
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Example
G =GL(3, R) acting on vectors
Let x be the first standard basis vector. Then Gx
is the subgroup
of G of all invertible matrices of
1 ∗
the form
.
0 ∗
A.M. Cohen, H. Cuypers, H. Sterk
Algebra 2
September 25, 2006
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Groups
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Example
Conjugation
Let x∈G . Then the stabilizer of x in G under conjugaction is CG (x), the subgroup of G
of all elements g with g ·x=x·g . This subgroup
is called the centralizer of x in G . It coincides
with the centralizer of the set x. Observe that
CG (1)=G .
A.M. Cohen, H. Cuypers, H. Sterk
Algebra 2
September 25, 2006
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Groups
Stabilizers
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Example
G =Dn on the vertices of a regular n-gon
Let G be the group Dn acting on the n vertices
of a regular n-gon. See a previous example. Let
x be a vertex. Among the n rotations in G only
the identity fixes x. The only reflection fixing x
is the reflection in the axis through x and the
center of the n-gon. So in this case Gx consists
of two elements.
A.M. Cohen, H. Cuypers, H. Sterk
Algebra 2
September 25, 2006
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Remark
The notation Gx does not explicitly use f . But
the stabilizer does depend on it. For instance, if
G =Sym4 and x=(1, 2), then
Gx =1 if f =L, left multiplication (or R∗ ,
right inverse multiplication);
Gx ={1, (1, 2), (3, 4), (1, 2)(3, 4)} if f =C,
conjugation.
A.M. Cohen, H. Cuypers, H. Sterk
Algebra 2
September 25, 2006
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Groups
Stabilizers
Algebra
Interactive
Remark
The notation Gx does not explicitly use f . But
the stabilizer does depend on it. For instance, if
G =Sym4 and x=(1, 2), then
Gx =1 if f =L, left multiplication (or R∗ ,
right inverse multiplication);
Gx ={1, (1, 2), (3, 4), (1, 2)(3, 4)} if f =C,
conjugation.
A.M. Cohen, H. Cuypers, H. Sterk
Algebra 2
September 25, 2006
9/9
Groups
Stabilizers
Algebra
Interactive
Remark
The notation Gx does not explicitly use f . But
the stabilizer does depend on it. For instance, if
G =Sym4 and x=(1, 2), then
Gx =1 if f =L, left multiplication (or R∗ ,
right inverse multiplication);
Gx ={1, (1, 2), (3, 4), (1, 2)(3, 4)} if f =C,
conjugation.
A.M. Cohen, H. Cuypers, H. Sterk
Algebra 2
September 25, 2006
9/9