The Journal of the Indian Mathematical Society

F. W. Levi ¹

Affiliations
1 Tata Institute of Fundamental Research, Bombay, IN

Abstract

In 1913, E. Helly[i] has established: "When C₁, ..., C_m are convex domains in an n-dimensional Euclidean space and every set of n+I of these domains have a common point, then there exists a point which is common to all the domains." Recently[6] this theorem has obtained an enhanced importance because of the numerous applications which it affords.

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F. W. Levi ¹

Affiliations
1 Bombay, IN

Abstract

In a recent paper, the author has introduced the notion of rank of commutativity of a ring. This notion will be applied to skewfields in the present note. It will be shown that the degree of a skewfield S can be characterized without explicit reference to the centre of S or to a basis of S over its centre.

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F. W. Levi ¹

Affiliations
1 Calcutta, IN

Abstract

Given an additive Abelian group A, the invariance of a subgroup B of A for an endomorphism a of A is a dyadic relation _apB between endomorphisms and subgroups of A. From every dyadic relation, two reciprocal lattices of "closed" sets are derived*.

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F. W. Levi ¹

Affiliations
1 Calcutta, IN

Abstract

In the first of these notes, the conditions have been discussed for a mapping {m} - i.e. the representation of each element a of a group G by a^m to be an endomorphism. Denoting by E(G) the set of numbers m for which {m} is an endomorphism and by M = M(G) the smallest positive number out of E(G) which maps G on an Abelian subgroup, say A(G), then the elements of E(G) were shown to form full classes of residues (mod M) satisfying certain conditions.

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F. W. Levi ¹

Affiliations
1 Calcutta, IN

Abstract

Let {m} be the mapping by which every element a of a group G is represented by its mth power a^m, then {o} and {1} are endomorphisms for every group, whereas {- 1} and {2} are endomorphisms if and only if G is an Abelian group.

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F. W. Levi ¹

Affiliations
1 Calcutta, IN

Abstract

In general, groups are determined by the help of an operation, called "composition" by which to every ordered pair a, b of elements a third element ab = c is attached. There is however a second mode of representing group theory, where instead of an operation operating on a pair of elements, a 3-term relation is introduced. E.g. the equation abc = 1 can be replaced by the statement that the relation R(a, b, c) holds.

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F. W. Levi ¹

Affiliations
1 Calcutta, IN

Abstract

This note deals with groups which are formed by two groups, say A and B, using certain homomorphisms. The resulting group [A, B] will be shown to be commutative and to depend on A/C(A) and B/C(B) only. When the two factorgroups admit representation by finite bases^ a similar representation of [A, B] can be derived (§3). Some results can be generalized for arbitrary groups A and B. Applications of the groups [A, B] will be made in some of the subsequent notes.

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F. W. Levi ¹

Affiliations
1 Calcutta, IN

Abstract

Denote the commutator of two group elements a and b by

aba^-1b^-1=(a,b)                                                               (1)

then the associative law for the commutator operation is

 (a, b), c) = (a, (b,c)).                                                    (2)

If (2) is satisfied for every triplet of elements of a group, this group will be called an S group. If (2) is supposed to be satisfied whenever two of the elements a, b, c are equal (alternating law), then the group will be called an L-group.

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F. W. Levi ¹

Affiliations
1 Calcutta University, IN

Abstract

Let a group G be generated by elements a₁,..., a_m which are supposed to be ≠I, but not by a smaller number of elements of G, then this minimum number of generators will be denoted by

m(G).                                                           (I)

If the order of G is equal to one, put m(G) = 0. The invariant m(G) of a group G has not yet been much investigated. One may suppose that the minimum number of generators of a free product is equal to the sum of the corresponding numbers of the free factors, but this supposition (which is true for large classes of groups) has neither been confirmed by a general proof, nor been refuted by a counter-example.

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F. W. Levi ¹

Affiliations
1 Calcutta University, IN

Abstract

The problem of arranging 2n+l persons on n different days around a circular table in such a way that no person has the same neighbour on different days, can be considered also as a problem on graphs, and the main result published in the paper under consideration.

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F. W. Levi ¹

Affiliations
1 Calcutta University, IN

Abstract

In this paper, a general method is given of finding the commutatorgroup C(G) of a free product

G = A*B                    (1)

of two groups A and B having only the unit element 1 in common. The commutatorgroups C(A) and C(B) as well as the factorgroups A/C(A) and B/C(B) are supposed to be known.

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F. W. Levi ¹

Affiliations
1 Calcutta University, IN

Abstract

If the projective axioms of the space are satisfied, then Desargues' theorem holds. If on the other hand in a plane the projective two-dimensional axioms are satisfied and Desargues' theorem holds, then the plane can be embedded into a space in which the projective axioms hold. The proof of the first proposition is easy and elementary; the second proposition has been proved by D. Hilbert by an ingenious arithmetical method which has the highest importance for the investigation of the fundamentals of Geometry. In this paper an alternative proof will be given which is elementary and geometrical; it applies a simple idea of descriptive geometry. Although descriptive geometry has occasionally been considered from an axiomatic point of view, its methods have-as far as 1 know-never been utilised for axiomatic purposes. As Descriptive Geometry is rather neglected in India, this paper is written in such a manner that the reader may follow it without references to that branch of mathematics.

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F. W. Levi ¹

Affiliations
1 University of Calcutta, IN

Abstract

Two submodules A and A' of a (commutative) field F will be said to be inverse, if to every element a≠0 of A, there exists in A' the inverse element a' = a^-1, and conversely. If A' is the same module as A, then A is said to be self-inverse. The rational homogeneous functions of x, y, z of order m with coefficients from an arbitrary field K, e.g. form a module which is inverse to a module formed by homogeneous functions of order - m. If m≠0, the two moduls have no common element, besides zero; if m = 0, the module is a field and therefore self-inverse.

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