Group action

This article is about the mathematical concept. For the sociology term, see group action (sociology).

Given an equilateral triangle, the counterclockwise rotation by 120° around the center of the triangle "acts" on the set of vertices of the triangle by mapping every vertex to another one.

In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set and the symmetries of the object are described by the symmetry group of this set, which consists of bijective [[ X is a regular covering space of another topological space Y, then the action of the deck transformation group on X is properly discontinuous as well as being free. Every free, properly discontinuous action of a group G on a path-connected topological space X arises in this manner: the quotient map $X\mapsto X/G$ is a regular covering map, and the deck transformation group is the given action of G on X. Furthermore, if X is simply connected, the fundamental group of $X/G$ will be isomorphic to $G$. These results have been generalised in the book Topology and Groupoids referenced below to obtain the fundamental groupoid of the orbit space of a discontinuous action of discrete group on a Hausdorff space, as, under reasonable local conditions, the orbit groupoid of the fundamental groupoid of the space. This allows calculations such as the fundamental group of a symmetric square.

An action of a group G on a locally compact space X is cocompact if there exists a compact subset A of X such that GA = X. For a properly discontinuous action, cocompactness is equivalent to compactness of the quotient space X/G.

The action of G on X is said to be proper if the mapping G×X → X×X that sends $(g,x)\mapsto (gx,x)$ is a proper map.

Strongly continuous group action and smooth points

If $\alpha :G\times X\to X$ is an action of a topological group $G$ on another topological space $X$, one says that it is strongly continuous if for all $x\in X$, the map $g\mapsto \alpha _{g}x$ is continuous with respect to the respective topologies. Such an action induce an action on the space of continuous function on $X$ by $(\alpha _{g}f)(x)=f(\alpha _{g}^{-1}x)$.

The subspace of smooth points for the action $\alpha$ is the subspace of $X$ of points $x$ such that $g\mapsto \alpha _{g}(x)$ is smooth, i.e. it is 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 does not 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. A morphism between G-sets is then a natural transformation between the group action functors. 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.

See also

• Group with operators
• Monoid action
• Gain graph

References

• Aschbacher, Michael (2000), Finite Group Theory, Cambridge University Press, ISBN 978-0-521-78675-1, MR1777008
• Brown, Ronald (2006). Topology and groupoids, Booksurge PLC, ISBN 1-4196-2722-8.
• Categories and groupoids, P.J. Higgins, downloadable reprint of van Nostrand Notes in Mathematics, 1971, which deal with applications of groupoids in group theory and topology.
• Dummit, David (2003). Abstract Algebra ((3rd ed.) ed.). Wiley. ISBN 0-471-43334-9. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
• Rotman, Joseph (1995). An Introduction to the Theory of Groups. Graduate Texts in Mathematics 148 ((4th ed.) ed.). Springer-Verlag. ISBN 0-387-94285-8.

• Weisstein, Eric W. "Group Action". MathWorld.