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Let G be a group, then there exists a piecewise linear knot K^(n-2) in S^n for n>=5 with G=pi_1(S^n-K) iff G satisfies 1. G is finitely presentable, 2. The Abelianization of ...

If n>19, there exists a Poulet number between n and n^2. The theorem was proved in 1965.

The first group isomorphism theorem, also known as the fundamental homomorphism theorem, states that if phi:G->H is a group homomorphism, then Ker(phi)⊴G and ...

The Schröder-Bernstein theorem for numbers states that if n<=m<=n, then m=n. For sets, the theorem states that if there are injections of the set A into the set B and of B ...

The first theorem of Pappus states that the surface area S of a surface of revolution generated by the revolution of a curve about an external axis is equal to the product of ...

A theorem, also called the iteration theorem, that makes use of the lambda notation introduced by Church. Let phi_x^((k)) denote the recursive function of k variables with ...

Suppose f(x) is a function of x that is twice differentiable at a stationary point x_0. 1. If f^('')(x_0)>0, then f has a local minimum at x_0. 2. If f^('')(x_0)<0, then f ...

The Hopf invariant one theorem, sometimes also called Adams' theorem, is a deep theorem in homotopy theory which states that the only n-spheres which are H-spaces are S^0, ...

The fourth group isomorphism theorem, also called the lattice group isomorphism theorem, lets G be a group and let N⊴G, where N⊴G indicates that N is a normal subgroup of G. ...

A special case of Stokes' theorem in which F is a vector field and M is an oriented, compact embedded 2-manifold with boundary in R^3, and a generalization of Green's theorem ...