Herzog–Schönheim conjecture

In mathematics, the Herzog–Schönheim conjecture is a combinatorial problem in the area of group theory, posed by Marcel Herzog and Jochanan Schönheim in 1974.[1]

Let ${\displaystyle G}$ be a group, and let

${\displaystyle A=\{a_{1}G_{1},\ \ldots ,\ a_{k}G_{k}\}}$

be a finite system of left cosets of subgroups ${\displaystyle G_{1},\ldots ,G_{k}}$ of ${\displaystyle G}$.

Herzog and Schönheim conjectured that if ${\displaystyle A}$ forms a partition of ${\displaystyle G}$ with ${\displaystyle k>1}$, then the (finite) indices ${\displaystyle [G:G_{1}],\ldots ,[G:G_{k}]}$ cannot be distinct. In contrast, if repeated indices are allowed, then partitioning a group into cosets is easy: if ${\displaystyle H}$ is any subgroup of ${\displaystyle G}$ with index ${\displaystyle k=[G:H]<\infty }$ then ${\displaystyle G}$ can be partitioned into ${\displaystyle k}$ left cosets of ${\displaystyle H}$.

Subnormal subgroups

In 2004, Zhi-Wei Sun proved an extended version of the Herzog–Schönheim conjecture in the case where ${\displaystyle G_{1},\ldots ,G_{k}}$ are subnormal in ${\displaystyle G}$.[2] A basic lemma in Sun's proof states that if ${\displaystyle G_{1},\ldots ,G_{k}}$ are subnormal and of finite index in ${\displaystyle G}$, then

${\displaystyle {\bigg [}G:\bigcap _{i=1}^{k}G_{i}{\bigg ]}\ {\bigg |}\ \prod _{i=1}^{k}[G:G_{i}]}$

and hence

${\displaystyle P{\bigg (}{\bigg [}G:\bigcap _{i=1}^{k}G_{i}{\bigg ]}\ {\bigg )}=\bigcup _{i=1}^{k}P([G:G_{i}]),}$

where ${\displaystyle P(n)}$ denotes the set of prime divisors of ${\displaystyle n}$.

Mirsky–Newman theorem

When ${\displaystyle G}$ is the additive group ${\displaystyle \mathbb {Z} }$ of integers, the cosets of ${\displaystyle G}$ are the arithmetic progressions. In this case, the Herzog–Schönheim conjecture states that every covering system, a family of arithmetic progressions that together cover all the integers, must either cover some integers more than once or include at least one pair of progressions that have the same difference as each other. This result was conjectured in 1950 by Paul Erdős and proved soon thereafter by Leon Mirsky and Donald J. Newman. However, Mirsky and Newman never published their proof. The same proof was also found independently by Harold Davenport and Richard Rado.[3]

In 1970, a geometric coloring problem equivalent to the Mirsky–Newman theorem was given in the Soviet mathematical olympiad: suppose that the vertices of a regular polygon are colored in such a way that every color class itself forms the vertices of a regular polygon. Then, there exist two color classes that form congruent polygons.[3]

References

1. Herzog, M.; Schönheim, J. (1974), "Research problem No. 9", Canadian Mathematical Bulletin, 17: 150. As cited by Sun (2004).
2. Sun, Zhi-Wei (2004), "On the Herzog-Schönheim conjecture for uniform covers of groups", Journal of Algebra, 273 (1): 153–175, arXiv:math/0306099, doi:10.1016/S0021-8693(03)00526-X, MR 2032455.
3. Soifer, Alexander (2008), "Chapter 1. A story of colored polygons and arithmetic progressions", The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of its Creators, New York: Springer, pp. 1–9, ISBN 978-0-387-74640-1.