UA-AUG-AUMAT229 Pre-read

Section 17.1 Pre-read

A central philosophy of group theory is that a group should measure symmetry. So generally the structure of a group is best studied by pairing a group with an object on which the group elements ``act'' by way of symmetries. This object does not have to be a geometric object. Usually it is a set, and group elements act by permuting the set.

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Example 17.1.1. $D_{3}$ acts on an equilateral triangle.

We created the group $D_{3}$ as the group of symmetries of an equilateral triangle. But it is more common to work with elements of $D_{3}$ as permutations of the vertices of that triangle. That is, if we number the vertices of the triangle, then we usually treat $D_{3}$ as a subgroup of $S_{3},$ even if we don't explicitly say so. (In fact, as we saw, $D_{3}$ is isomorphic to $S_{3} .$ )

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Example 17.1.2. ${GL}_{n} (R)$ and its subgroups act on $R^{n}$ .

A matrix times a column vector results in another column vector, so we can think of any group of invertible matrices as a group of geometric transformations of $R^{n} .$ That is, we can think of invertible matrices as elements of $S_{R^{n}} .$

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So, just as we did when we explored Cayley's Theorem, we want to relate a given group

G

to some subgroup of

S_{X}

for some set

X,

However, in Cayley's Theorem we took a very particular approach for a very particular purpose; now we will be much more general.

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Definition 17.1.3. Group action.

A homomorphism from a group $G$ to the group $S_{X}$ of permutations on some set $X .$

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If we have such a homomorphism

φ : G \to S_{X},

then for

g

G

the image

φ (g)

is an element of

S_{X},

so it is a function itself; in particular, each

φ (g) : X \to X

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is a bijection. It is usually too clumsy to use notation like

φ (g) (x)

to mean the result of interpreting an element

g

G

as a permutation of the set

X

and then applying that permutation to the specific element

x

X,

so we usually just interpret

g

itself as a function on

X

and write

g (x)

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instead. (Another common notation is to use multiplicative notation in analogy with the matrices-acting-on-vectors example and writing

g \cdot x,

but we will avoid this notation to avoid confusion with the group operation in

G .

)

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Here are two important concepts related to group actions. Suppose group

G

acts on set

X,

and

x

is some specific element in

X .

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Definition 17.1.4. Stabilizer of set element $x$ .

Write $G_{x}$ or ${Stab}_{G} (x)$ to mean the collection of those group elements $g$ so that $g (x) = x .$

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Definition 17.1.5. Orbit of set element $x$ .

Write $G (x)$ or $O_{x}$ to mean the collection of all possible results of applying the group action on $x .$ That is, an element $y$ in $X$ is in the orbit $G (x)$ precisely when there is at least one $g$ in $G$ so that $y = g (x) .$

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Warning 17.1.6. Keep it straight!

A stabilizer is a collection of group elements from $G,$ while an orbit is a collection of set elements from $X .$

Section 17.1 Pre-read

Example 17.1.1. D3 acts on an equilateral triangle.

Example 17.1.2. GLn(R) and its subgroups act on Rn.