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Let G be a finitely generated nilpotent group with lower central series
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Let and G
* Let H be a group, and let G be the direct product H × H. Then the subgroups
Suppose that in a mathematical language L, it is possible to enumerate all of the defined numbers in L. Let this enumeration be defined by the function G: W → R, where G ( n ) is the real number described by the nth description in the sequence.
Let G denote the set of bijective functions over A that preserve the partition structure of A: ∀ x ∈ A ∀ g ∈ G ( g ( x ) ∈ ).
Moving to groups in general, let H be a subgroup of some group G. Let ~ be an equivalence relation on G, such that a ~ b ↔ ( ab < sup >− 1 </ sup > ∈ H ).
Let f and g be any two elements of G. By virtue of the definition of G, = and =, so that =.
Let G be a set and let "~" denote an equivalence relation over G. Then we can form a groupoid representing this equivalence relation as follows.
Let a, b, and c be elements of G. Then:
Let E be the intersection of the diagonals, and let F be on side DA and G be on side BC such that FEG is parallel to AB and CD.
Let R be a ring and G be a monoid.
Let this point be called G. Point G is also the midpoint of line segment FQ:
Let F and G be a pair of adjoint functors with unit η and co-unit ε ( see the article on adjoint functors for the definitions ).
Let ( G ,.
Let G and H be groups, and let φ: G → H be a homomorphism.
Let G be a group.
Let S be a subgroup of G, and let N be a normal subgroup of G. Then:
Let G be a group.
Let N and K be normal subgroups of G, with
Let and be
Let the open enemy to it be regarded as a Pandora with her box opened ; ;
Let every policeman and park guard keep his eye on John and Jane Doe, lest one piece of bread be placed undetected and one bird survive.
`` Let him be now ''!!
Let us assume that it would be possible for an enemy to create an aerosol of the causative agent of epidemic typhus ( Rickettsia prowazwki ) over City A and that a large number of cases of typhus fever resulted therefrom.
Let T be a linear operator on the finite-dimensional vector space V over the field F.
Let p be the minimal polynomial for T, Af, where the Af, are distinct irreducible monic polynomials over F and the Af are positive integers.
Let Af be the null space of Af.
Let N be a linear operator on the vector space V.
Let T be a linear operator on the finite-dimensional vector space V over the field F.
Let V be a finite-dimensional vector space over an algebraically closed field F, e.g., the field of complex numbers.
Let N be a positive integer and let V be the space of all N times continuously differentiable functions F on the real line which satisfy the differential equation Af where Af are some fixed constants.
Let Q be a nonsingular quadric surface bearing reguli Af and Af, and let **zg be a Af curve of order K on Q.
Let us take a set of circumstances in which I happen to be interested on the legislative side and in which I think every one of us might naturally make such a statement.
Let the state of the stream leaving stage R be denoted by a vector Af and the operating variables of stage R by Af.
Let this be denoted by Af.
Let it be granted then that the theological differences in this area between Protestants and Roman Catholics appear to be irreconcilable.
Let not your heart be troubled, neither let it be afraid ''.
The same God who called this world into being when He said: `` Let there be light ''!!
For those who put their trust in Him He still says every day again: `` Let there be light ''!!
Let us therefore put first things first, and make sure of preserving the human race at whatever the temporary price may be ''.
Let her out, let her out -- that would be the solution, wouldn't it??
Let and finitely
Let S be the group of all permutations of N, the natural numbers, that fixes all but finitely many numbers then:
Let R be a Noetherian ring and let I be an ideal of R. Then I may be written as the intersection of finitely many primary ideals with distinct radicals ; that is:
Let us briefly recall the structure theory in the case of a finitely generated module over a PID.
Let be a finitely generated field extension of a field.
Let P be a finitely generated projective module over a commutative ring R and X be the spectrum of R. The rank of P at a prime ideal in X is the rank of the free-module.
Let k be a field and A finitely generated k-algebra.
Let A ⊆ R → S be homomorphisms where R is not necessarily local ( one can reduce to that case however ), with A, S regular and R finitely generated as an A-module.
Let G be a finitely generated group.
Let G be a finitely generated torsionfree nilpotent group.
Let k be a field and M a finitely generated module over the polynomial ring
* Let M be a finitely generated module over a local ring.
* Let R be a local ring and M a finitely generated module over R. Then the projective dimension of M over R is equal to the length of every minimal free resolution of M. Moreover, the projective dimension is equal to the global dimension of M, which is by definition the smallest integer i ≥ 0 such that
: Nakayama's lemma: Let U be a finitely generated right module over a ring R. If U is a non-zero module, then U · J ( R ) is a proper submodule of U.
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