Part 4:  GROUP ACTIONS

1: What is a group action? 

 There is a common thread in the ideas of non-singular linear transformations acting on a vector space and D4 acting on the corners, or the diagonals of a square.

 In each case we have a group G a set X and a mapping q: G x X -> X, denoted (g,x) |-> xg , satisfying:

     
  1. x1 = x 
  2. (xg)h = x(gh)
for all x in X and for all g, h in G.

 Such a mapping is called a group G acting on a set X.

 Here is another way of looking at it: Suppose q is an action of G on X. Define a mapping y: G -> SymX, the group of permutations of X by gy: x |-> xg. First it must be checked that gy in SymX. This is true because it is a mapping from X to X which is 1 - 1 and onto because it has an inverse g-1.y. Now you can check that y is a homomorphism, because for all g and h in G (gh)y does the same to any x in X as 
(g y)(hy).

 This shows that every group action of G on X gives rise to a homomorphism y: G -> SymX. Conversely, given a homomorphism y: G -> SymX, define an action q of G on X by (g,x) |-> x g. It is straightforward to check that q satisfies the axioms of a group action. Thus we have the complete logical equivalence of a group G acting on a set X and a homomorphism of G into SymX.

 Group actions can be used for studying the set X, especially if it has some other structure, like a graph, vector space or module, but its most important use in this course will be for studying the structure of the group G. Another name for a homomorphism y: G -> SymX is a permutation representation of G in X or in SymX.

 

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2: Linear representations

 By the way, a homomorphism of a group G into GL(V) for some vector space V is called a linear representation of G. Examples are the representations of the symmetry groups of the square and cube by non-singular matrices with entries 0, 1 and -1 we looked at earlier.

 The kernel of a representation of G or action of G on X is just the kernel of y, i.e. the set 

{g in G: for all x in X, xg = x}.

A representation or action is called faithful if it has kernel zero, and transitive if for all x and y in X there is some g in G such that xg = y. Note that this is not the same as saying that y is onto. It is however equivalent to the following statement: let x be a fixed element of X. Then for all y in X there is a g in G such that y = xg.

 

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3: Orbits of group actions 

 If x in X, the orbit of x is the set {xg: g in G}. At first sight, it looks like the orbit of x is the same size as G, but this is false in general, because there may be many g and h in G with xg = xh. What we can say however is:

Lemma 4.1 The set of orbits in X of G is a partition of X. 

Proof It is enough to prove that being in the same orbit is an eqivalence relation on X.

     
  1. Since for all x in X, x = x1, the relation is reflexive.
  2. Let y be in the orbit of x, so y = xg, say. Then x = yg-1, so x is in the orbit of y. Thus the relation is symmetric.
  3. Let y be in the orbit of x, and z in the orbit of y. Then y = xg and z = yh. Hence z = xgh so z is in the orbit of x. Thus the relation is transitive.
The set of elements x of X whose orbit is just {x} is called the fixed set of the action. 

Notice that if G acts on X and H is a subgroup of G, then H also acts on X and the orbits of X under H are subsets of the orbits of X under G, and thus induce a finer partition of X.

 On the other hand, we have:

Lemma 4.2 Let G be a group acting on a set X, and let Y be a subset of X. Then the action of G on X induces an action of G on Y if and only if Y is a union of orbits of X under G.

 

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4.Fixed groups of actions 

 Now suppose that x is an element of X and let 
     Gx = {a in G: x a = x}.

 Lemma 4.3 Gx is a subgroup of G. Moreover there is a 1-1 correspondence between the orbit of x under the action of G and the cosets of Gx in G.

 Proof. Denote the orbit of x by Orb(x) and the set of cosets by [G:Gx]. Define F: Orb(x) -> [G:Gx] by 
    F: x a |-> Gxa.

 F is well-defined, for if x a = x b, then a.b-1 fixes x. Hence a.b-1 in Gx, so Gxa  = Gxb.

 F is 1-1, for if Gxa = Gxb  then a.b-1 in Gx, so 
x = x a.b-1 and hence x a = x b.

 F is onto, for if Gxa  is a coset, it is the image under F of x a.

 Corollary 7.4 If G is a finite group acting on any set X, then the orbits of elements of X are finite. Moreover, their length divides the order of G.

 

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5: Examples 

 It's time for some examples. 
  1. The dihedral group D4 acts faithfully on the set of corners of the square. The corresponding homomorphism D4 -> Sym4 satisfies:
    •  I -> (1)
    • r -> (1234)
    • r2 -> (13)(24)
    • r3 -> (1432)
    • f -> (12)(34)
    • r.f -> (24)
    • r2.f -> (14)(23)
    • r3.f -> (13).
    Its image is a non-normal subgroup of order 8 whose conjugates in Sym4 are the other two isomorphic subgroups obtained by starting with the permutations (1324) and (1243).
  2. The dihedral group D4 acts on the set {/,\}of diagonals. There are two permutations of this set, say I, the identity, and J which swaps them. The corresponding homomorphism D4 -> Sym{/,\} satisfies:
    •  I -> I
    • r -> J
    • r2 -> I
    • r3 -> J
    • f -> J
    • r.f -> I
    • r2.f -> J
    • r3.f -> I.
    Thus the kernel is the normal subgroup of order 4 generated by r2 and r.f, which is isomorphic to C2 X C2. 
  3. The octahedral group O3 acts faithfully on the set of long diagonals of the cube, as we have already seen.
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 Last update: 10 October, 1999

 Author: Phill Schultz, schultz@maths.uwa.edu.au