Group action Totally Explained

Everything about Group Action totally explained

In mathematics, a symmetry group describes all symmetries of objects. This is formalized by the notion of a group action: every element of the group "acts" like a bijective map (or "symmetry") on some set. In this case, the group is also called a permutation group (especially if the set is finite or not a vector space) or transformation group (especially if the set is a vector space and the group acts like linear transformations of the set). A permutation representation of a group G is a representation of G as a group of permutations of the set (usually if the set is finite), and may be described as a group representation of G by permutation matrices, and is usually considered in the finite-dimensional case—it is the same as a group action of G on an ordered basis of a vector space.

Definition

If G is a group and X is a set, then a (left) group action of G on X is a binary function ť

G 	imes X 	o X,

denoted ť

(g,x)mapsto gcdot x,

which satisfies the following two axioms:

(gh)·x = g·(h·x) for all g, h in G and x in X
e·x = x for every x in X (where e denotes the identity element of G)

The set X is called a (left) G-set. The group G is said to act on X (on the left).
From these two axioms, it follows that for every g in G, the function which maps x in X to g·x is a bijective map from X to X. Therefore, one may alternatively define a group action of G on X as a group homomorphism from G into the symmetric group S_X.
In complete analogy, one can define a right group action of G on X as a function X × G → X by the two axioms:

x·(gh) = (x·g)·h

x·e = x The difference between left and right actions is in the order in which a product like gh acts on x. For a left action h acts first and is followed by g, while for a right action g acts first and is followed by h. From a right action a left action can be constructed by composing with the inverse operation on the group. If r is a right action, then ť

l : G 	imes M 	o M : (g, m) mapsto r(m, g^x)

.
The space of smooth vector for the action

alpha

is the subspace of

A

of elements

a

such that

xmapstoalpha_x(a)

is smooth, for example it's continuous and all derivatives are continuous.

Generalizations

One can also consider actions of monoids on sets, by using the same two axioms as above. This doesn't define bijective maps and equivalence relations however.
   Instead of actions on sets, one can define actions of groups and monoids on objects of an arbitrary category: start with an object X of some category, and then define an action on X as a monoid homomorphism into the monoid of endomorphisms of X. If X has an underlying set, then all definitions and facts stated above can be carried over. For example, if we take the category of vector spaces, we obtain group representations in this fashion.
   One can view a group G as a category with a single object in which every morphism is invertible. A group action is then nothing but a functor from G to the category of sets, and a group representation is a functor from G to the category of vector spaces. In analogy, an action of a groupoid is a functor from the groupoid to the category of sets or to some other category.
   Without using the language of categories, one can extend the notion of a group action on a set X by studying as well its induced action on the power set of X. This is useful, for instance, in studying the action of the large Mathieu group on a 24-set and in studying symmetry in certain models of finite geometries.

Further Information

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