Math and science::Algebra::Aluffi

Groups. Group action.

Consider a group and a target object, such as a set. A group actions associate each group element with an automorphism of a target object, such as a set.

Group action

Let $G$ be a group with operation $m_{G} : G \times G \to G$ , and let $A$ and object. A function $σ : G \to Auto (A)$ is a group action iff:

the identity of $G$ maps to the identity of $A$ .
either:
1. for any $g, h \in G$ , $σ (m_{G} (g, h)) =$ $σ (g) \circ σ (h)$ [left-action]
2. for any $g, h \in G$ , $σ (m_{G} (g, h)) =$ $σ (h) \circ σ (h)$ [right-action]

Why are there two types of actions?

Composition order

The need for two action types arises from the two ways of mapping the parameters that go into $m_{G} (_,_)$ to the "inputs" of function composition $_\circ_$ .

Consider $σ$ as a group homomorphism from $(G, +)$ to a group $(F_{A}, @)$ where the elements of $F_{A}$ are automorphisms of $A$ . The operation $@$ can take one of two forms:

$@$ is standard function composition: $f @ g (x) = f (g (x))$
$@$ is the opposite of standard function composition: $f @ g (x) = g (f (x))$

The 1^st operation corresponds to a left-action and the 2^nd operation to a right-action. If, for the 2^nd operation, we write functions on the right of their arguments:

(x) f @ g = ((x) f) g

Then we can see how the terms left and right arise. This is covered more below.

Compact notation

Let $g, h \in G$ and $x \in A$ . For a left-action (second parameter goes first), typical notation will express the second requirement in the definition above as:

σ (m_{G} (g, h)) (x) = σ (g) \circ σ (h) (x)

to:

(a h) . x = a . (h . x)

The dot is often dropped to arrive at:

(a h) x = a (h x)

And because these two terms are equal, the brackets are often dropped.

The requirement for the right-action is similarly shortened to:

x . (a h) = (x . a) . h

and

x (a h) = (x a) h

Comparing the two versions, we see that left-actions act like functions that are written on the left of their arguments, while right-actions are like functions that are written on the right of their arguments.

Datum of a group action

A datum of a group action can be considered a table with a single column:

Table for $σ : G \to Auto (A)$

$e_{G}$	$i_{A}$
a	$ϕ_{j}$
b	$ϕ_{k}$
c	$ϕ_{l}$
d	$ϕ_{x}$
f	$ϕ_{y}$
g	$ϕ_{z}$
h	$ϕ_{w}$

Alternatively, if the data of the automorphisms is unwrapped, the datum can be considered a 2D table. For the case where the target object is a set, it might look like the table below.

Table for $ρ : G \to A$
	$x_{1}$	$x_{2}$	$x_{3}$	$x_{4}$	$x_{5}$	$x_{6}$	$x_{7}$	$x_{8}$	$x_{9}$	$x_{10}$	$x_{11}$	$x_{12}$	$. . .$	$x_{n}$
$e_{G}$	$x_{1}$	$x_{2}$	$x_{3}$	$x_{4}$	$x_{5}$	$x_{6}$	$x_{7}$	$x_{8}$	$x_{9}$	$x_{10}$	$x_{11}$	$x_{12}$	$. . .$	$x_{n}$
a	$x_{17}$	$x_{41}$	$x_{12}$	$x_{4}$	$x_{10}$	$x_{14}$	$x_{2}$	$x_{8}$	$x_{21}$	$x_{19}$	$x_{15}$	$x_{14}$	$. . .$	$x_{?}$
b
c
d
f
g
h

Example

Target is a set

When the target is a set, $A$ , the set of automorphisms is the set of permutations of the set—the symmetry group, $S_{A}$ (or is it an isomorphism to it?).