* Any vector space V endowed with the identically zero Lie bracket becomes a Lie algebra.

Any one-dimensional Lie algebra over a field is abelian, by the antisymmetry of the Lie bracket.

* Any Lie group G defines an associated real Lie algebra.

* Any topologically closed subgroup of a Lie group is a Lie group.

* Any simply connected solvable Lie group is isomorphic to a closed subgroup of the group of invertible upper triangular matrices of some rank, and any finite dimensional irreducible representation of such a group is 1 dimensional.

* Any simply connected nilpotent Lie group is isomorphic to a closed subgroup of the group of invertible upper triangular matrices with 1's on the diagonal of some rank, and any finite dimensional irreducible representation of such a group is 1 dimensional.

Any Lie group G can be decomposed into discrete, simple, and abelian groups in a canonical way as follows.

# Any tangent vector at the identity of a Lie group can be extended to a left invariant vector field by left translating the tangent vector to other points of the manifold.

A weight on a Lie algebra g over a field F is a linear map λ: g → F with λ ( y )= 0 for all x, y in g. Any weight on a Lie algebra g vanishes on the derived algebra and hence descends to a weight on the abelian Lie algebra g /.

* Any foliation gives a Lie groupoid.

Any smooth function on a symplectic manifold gives rise, by definition, to a Hamiltonian vector field and the set of all such form a subalgebra of the Lie Algebra of symplectic vector fields.

Any associative algebra A over the field K becomes a Lie algebra over K with the Lie bracket:

* Any Lie group with an infinite group of components G / G < sup > o </ sup > cannot be realized as an algebraic group ( see identity component ).

Any n-dimensional formal group law gives an n dimensional Lie algebra over the ring R, defined in terms of the quadratic part F < sub > 2 </ sub > of the formal group law.

Any finite-dimensional irreducible representation of a semisimple Lie group or Lie algebra can be constructed from the fundamental representations by a procedure due to Élie Cartan.

* Any member of the genus Eunectes, a group of large, aquatic snakes found in South America

Any normal subgroup has a corresponding quotient group, formed from the larger group by eliminating the distinction between elements of the subgroup.

Any formal or informal group — a family, a church, a club, a business, a trade union — may be said to have government.

" Any attempt to organize the group ... under a single authority would eliminate their independent initiatives, and thus reduce their joint effectiveness to that of the single person directing them from the centre.

Any symmetry group whose elements have a common fixed point, which is true for all finite symmetry groups and also for the symmetry groups of bounded figures, can be represented as a subgroup of orthogonal group O ( n ) by choosing the origin to be a fixed point.

However, in the case of a finitely presented group we know that not all the generators can be trivial ( Any individual generator could be, of course ).

* Any subgroup of a free group is free.

Any group of four students may run for office, but there must always be four students.

Any group that managed to find ways of reasoning effectively would reap benefits for all its members, increasing their fitness.

Any time during a triad conversation, group members can switch seats and one of the co-pilots can sit in the pilot ’ s seat.

Any group can be seen as a category with a single object in which every morphism is invertible ( for every morphism f there is a morphism g that is both left and right inverse to f under composition ) by viewing the group as acting on itself by left multiplication.

* Any expression formed using any combination of the basic arithmetic operations and extraction of nth roots gives an algebraic number.

Any accelerating electric charge, and therefore any changing electric current, gives rise to an electromagnetic wave that propagates at very high speed outside the surface of the conductor.

Any second-countable space is separable: if is a countable base, choosing any gives a countable dense subset.

* Properly with a generic referent: " Any cow gives milk.

Any meromorphic function f gives rise to a divisor denoted ( f ) defined as

* Any principal bundle with structure group G gives a groupoid, namely over M, where G acts on the pairs componentwise.

Any point x gives rise to a continuous function p < sub > x </ sub > from the one element topological space 1 ( all subsets of which are open ) to the space X by defining p < sub > x </ sub >( 1 ) = x. Conversely, any function from 1 to X clearly determines one point: the element that it " points " to.

Any ξ ∈ gives rise to a canonical vertical vector field X < sub > ξ </ sub > by taking the derivative of the right action of the 1-parameter subgroup of H associated to ξ.

Any prime number p gives rise to an ideal pO < sub > K </ sub > in the ring of integers O < sub > K </ sub > of a quadratic field K.

Any surface that intersects the wire has current I passing through it so Ampère's law gives the correct magnetic field.

Any ring homomorphism A → B gives a map K < sub > 0 </ sub >( A ) → K < sub > 0 </ sub >( B ) by mapping ( the class of ) a projective A-module M to M ⊗< sub > A </ sub > B, making K < sub > 0 </ sub > a covariant functor.

:( 1 ) Any person who manufactures, sells or hires or offers for sale or hire, or exposes or has in his possession for the purpose of sale or hire or lends or gives to any other person —

Any kind of connection on a manifold gives rise, through its parallel transport maps, to some notion of holonomy.

Any skew semistandard tableau of shape / with positive integer entries gives rise to a sequence of partitions ( or Young diagrams ), by starting with, and taking for the partition places further in the sequence the one whose diagram is obtained from that of by adding all the boxes that contain a value ≤ in ; this partition eventually becomes equal to.

Any attempt to drastically increase this range ( by increasing the angular extent of the true radius lobe nose region ) soon gives rise to problems with excessive rates of valve acceleration etc.

Any nontrivial line arrangement on RP < sup > 2 </ sup > defines a graph in which each face is bounded by at least three edges, and each edge bounds two faces ; so, double counting gives the additional inequality F ≤ 2E / 3.

Let now v and w be two vertices in G. Any spanning tree contains precisely one simple path between v and w. Taking this path in the uniform spanning tree gives a random simple path.

Any symmetric space gives an involutory quandle, where is the result of ' reflecting through '.

Any topological space X gives rise to a category where the objects are the points of X and the morphisms are the homotopy classes of paths.

In his letter, Polding gives Wardell a completely free hand in the design, saying " Any plan, any style, anything that is beautiful and grand, to the extent of our power.

Any premetric gives rise to a preclosure operator cl as follows: