Let G be a subgroup of S_n. Then the sequence [1, ..., n-1] is a basis for G. If G = S_n, then we cannot replace the basis by a smaller one.

If G = A_n, then [1, ..., n - 2] is a basis. For, the only nontrivial permutation in S_n stabilizing each of the elements 1, ..., n - 2 is the transposition (n - 1,n). But this is element is odd and so does not belong to A_n.

Let V be a vector space of dimension n. Consider G = GL(V) acting on the vectors of the vector space V.

If v₁, ..., v_n is a basis of V, then [v₁, ..., v_n] is also a basis of G acting on V, for a linear transformation fixing a basis of V is the identity.

Consider G = D_n acting on the n vertices of the n-gon. Any two vertices that are not opposite form a basis for G.