In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Herzog–Schönheim conjecture is a combinatorial problem in the area of group theory

Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

.

Let be a group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

, and let


be a finite system of left cosets

Coset

In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, thenA coset is a left or right coset of some subgroup in G...

 of subgroup

Subgroup

In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...

s
of .

In 1974 M. Herzog and J. Schönheim conjectured
that if forms a partition

Partition of a set

In mathematics, a partition of a set X is a division of X into non-overlapping and non-empty "parts" or "blocks" or "cells" that cover all of X...

 of
with ,
then the (finite) indices cannot be distinct. In contrast, if is a subgroup of
with index

Index of a subgroup

In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" of H that fill up G. For example, if H has index 2 in G, then intuitively "half" of the elements of G lie in H...

  then can be partitioned into left cosets of .

When is the additive group of integers, this is a conjecture of Paul Erdős

Paul Erdos

Paul Erdős was a Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory...

 confirmed by H. Davenport

Harold Davenport

Harold Davenport FRS was an English mathematician, known for his extensive work in number theory.-Early life:...

, L. Mirsky, D. Newman and R. Rado

Richard Rado

Richard Rado FRS was a Jewish German mathematician. He earned two Ph.D.s: in 1933 from the University of Berlin, and in 1935 from the University of Cambridge. He was interviewed in Berlin by Lord Cherwell for a scholarship given by the chemist Sir Robert Mond which provided financial support to...

 independently. Observe that


are pairwise disjoint with the indices distinct, but they do not cover multiples of .

In 2004 Zhi-Wei Sun

Zhi-Wei Sun

Zhi-Wei Sun is a Chinese mathematician, working primarily on number theory, combinatorics, and group theory. Currently he works as a professor in Nanjing University....

 proved an extended version
of the Herzog–Schönheim conjecture in the case where are subnormal

Subnormal subgroup

In mathematics, in the field of group theory, a subgroup H of a given group G is a subnormal subgroup of G if there is a finite chain of subgroups of the group, each one normal in the next, beginning at H and ending at G....

 in
http://arxiv.org/abs/math/0306099. A basic lemma in Sun's proof states that if are subnormal and of finite index in , then


and hence


where denotes the set of prime

Prime number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

divisor

Divisor

In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...

s of .