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In the plane, there are 17 lattice groups, eight of which are pure translation. In R^3, there are 32 point groups and 230 space groups. In R^4, there are 4783 space lattice ...

The Lebesgue identity is the algebraic identity (Nagell 1951, pp. 194-195).

For a group G, consider a subgroup H with elements h_i and an element x of G not in H, then xh_i for i=1, 2, ... constitute the left coset of the subgroup H with respect to x.

In a noncommutative ring R, a left ideal is a subset I which is an additive subgroup of R and such that for all r in R and all a in I, ra in I. A left ideal of R can be ...

Let H be a subgroup of G. A subset T of elements of G is called a left transversal of H if T contains exactly one element of each left coset of H.

A Lie groupoid over B is a groupoid G for which G and B are differentiable manifolds and alpha,beta and multiplication are differentiable maps. Furthermore, the derivatives ...

Any linear system of point-groups on a curve with only ordinary singularities may be cut by adjoint curves.

A subset M of a Hilbert space H is a linear manifold if it is closed under addition of vectors and scalar multiplication.

Two curves phi and psi satisfying phi+psi=0 are said to be linearly dependent. Similarly, n curves phi_i, i=1, ..., n are said to be linearly dependent if sum_(i=1)^nphi_i=0.

This is proven in Rademacher and Toeplitz (1957).