Proper action

In mathematics, an action of a group is a formal way of interpreting the manner in which the elements of the group correspond to transformations of some space in a way that preserves the structure of that space. Common examples of spaces that groups act on are sets, vector spaces, and topological spaces. Actions of groups on vector spaces are called representations of the group.

When there is a natural correspondence between the set of group elements and the set of space transformations, a group can be interpreted as acting on the space in a canonical way. For example, the symmetric group of a finite set consists of all bijective transformations of that set; thus, applying any element of the permutation group to an element of the set will produce another (not necessarily distinct) element of the set. More generally, symmetry groups such as the homeomorphism group of a topological space or the general linear group of a vector space, as well as their subgroups, also admit canonical actions. For other groups, an interpretation of the group in terms of an action may have to be specified, either because the group does not act canonically on any space or because the canonical action is not the action of interest. For example, we can specify an action of the two-element cyclic group ${\displaystyle \mathrm {C} _{2}=\{0,1\}}$ on the finite set ${\displaystyle \{a,b,c\}}$ by specifying that 0 (the identity element) sends ${\displaystyle a\mapsto a,b\mapsto b,c\mapsto c}$, and that 1 sends ${\displaystyle a\mapsto b,b\mapsto a,c\mapsto c}$. This action is not canonical.

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