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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 \( \sigma : G \to \operatorname{Auto}(A) \) is a group action iff:

  1. the identity of \( G \) maps to the identity of \( A \).
  2. either:
    1. for any \( g, h \in G \), \( \sigma(\; m_G(g, h)\;) \; = \; \) \( \sigma(g) \circ \sigma(h) \;\; \) [left-action]
    2. for any \( g, h \in G \), \( \sigma(\; m_G(g, h)\;) \; = \; \) \( \sigma(h) \circ \sigma(h) \;\; \) [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 \( \sigma \) as a group homomorphism from \( (G, +) \) to a group \( (F_A, \operatorname{@} ) \) where the elements of \( F_A \) are automorphisms of \( A \). The operation \( \operatorname{@} \) can take one of two forms:

  1. \( @ \) is standard function composition: \( f \operatorname{@} g(x) = f(\,g(x)\,) \)
  2. \( @ \) is the opposite of standard function composition: \( f \operatorname{@} g(x) = g(\,f(x)\,) \)

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

\[ (x) f \operatorname{@} g = (\, (x)f \; )g \]

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:

\[ \sigma(\; m_G(g, h) \;)(x) = \sigma(g) \circ \sigma(h)(x) \]

to:

\[ (ah).x = a.(h.x) \]

The dot is often dropped to arrive at:

\[ (ah)x = a(hx) \]

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

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

\[ x.(ah) = (x.a).h \]

and

\[ x(ah) = (xa)h \]

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 \( \sigma : G \to \operatorname{Auto}(A) \)
\( e_G \) \( i_A \)
a \( \phi_j \)
b \( \phi_k \)
c \( \phi_l \)
d \( \phi_x \)
f \( \phi_y \)
g \( \phi_z \)
h
 \( \phi_w \)

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 \( \rho : G \to A \)
\( x_1 \) \( x_2 \) \( x_3 \) \( x_4 \) \( x_5 \) \( x_6 \) \( x_7 \) \( x_8 \) \( x_9 \) \( x_{10} \) \( x_{11} \) \( x_{12} \) \( ... \) \( x_n \)
\( e_G \) \( x_1 \) \( x_2 \) \( x_3 \) \( x_4 \) \( x_5 \) \( x_6 \) \( x_7 \) \( x_8 \) \( x_9 \) \( x_{10} \) \( x_{11} \) \( x_{12} \) \( ... \) \( x_n \)
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?).