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A nonempty set $A$ equipped with a group action of the additive group of a vector space $V$, such that for every $x,y\in A$ there is a unique $v\in V$ with $x+v=y$. There is no distinguished origin. Choosing one identifies $A$ with $V$; changing it gives a translation.

A block system for a group action is a partition of the acted-on set such that every group element permutes its parts. For a transitive action, a nontrivial block system has more than one part and more than one element per part. The full symmetric group preserves no such partition: a transposition exchanging one point in each of two parts breaks it.

For a group action on $X$ and a contractible free $G$-space $EG$, the diagonal orbit space $EG{\times }_{G}X$ maps to the classifying space $BG$ with fibre $X$. It defines equivariant cohomology. When $X\to X/G$ is a principal $G$-bundle, the map $EG{\times }_{G}X\to X/G$ has contractible fibre $EG$ and gives a homotopy equivalence for spaces of CW type. Freeness without the bundle hypotheses should not be substituted for this assertion for arbitrary topological groups.

The coadjoint action of a Lie group on the dual space of its Lie algebra is $({\mathrm{Ad}}_g^{\ast }\eta )(X)=\eta ({\mathrm{Ad}}_{g^{-1}}X)$. The inverse in this formula makes it a left group action.

An isometric group action on a metric space is cobounded if some orbit is within a fixed distance $R$ of every point. Equivalently, translates of one bounded ball cover the space. Coboundedness does not require finite stabilizers, a proper space, or a finitely generated group. Every action on a point is cobounded.

The full symmetry group of a cube acts transitively and faithfully on its twelve edges. For vertices $(\pm 1,\pm 1,\pm 1)$, its signed permutation matrices act transitively on the edges. The stabilizer subgroup of $\{(1,1,z):-1\le z\le 1\}$ consists of independently swapping $x,y$ and reversing $z$, and is a Klein four-group. The orbit-stabilizer theorem gives order $12\cdot 4=48$. Fixing every edge fixes every vertex, as each vertex is the unique intersection of its three incident edges, so the group action is faithful. The resulting subgroup of $S_{12}$ has index $12!/48$ and is not normal: central inversion has cycle type $2^6$, whose conjugacy class in $S_{12}$ has $12!/(2^{6}6!)=10395>48$ elements.

The Borel version of equivariant cohomology is the ordinary cohomology of the Borel construction. It retains information about both the space and its group action. For a point it is the cohomology of the classifying space; for a free action with the usual bundle hypotheses it is the cohomology of the orbit space.

If $G$ has $d$ generators, there are at most $(n!{)}^d$ group homomorphisms $G\to S_n$. Each subgroup of index $n$ is a point stabilizer in a transitive coset group action, so there are at most $n(n!{)}^d$ such subgroups. Consequently finitely many subgroups have index at most any fixed bound. Intersecting them gives a finite-index characteristic subgroup, useful in proving residual finiteness of semidirect products.

A fixed point of a group action is an element fixed by every group element. The set $A^G=\{a:g\cdot a=a\text{ for all }g\in G\}$ is the categorical limit of the action viewed as a functor from the one-object category associated with $G$ to the Category of sets.

Invariant theory studies functions and tensors unchanged by a group action. For a linear representation, the polynomial invariant ring records invariant polynomial functions on the representation space. Tensor invariants and commuting algebra actions connect it with Schur–Weyl duality.

A Lie group action is a smooth group action of a Lie group on a smooth manifold. Differentiating the action gives an infinitesimal action of its Lie algebra by vector fields.