Herzog–Schönheim conjecture explained

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

G be a group, and letA = \{a _{1G} _{1, \ldots, a} _{kG} _{k\}}

be a finite system of left cosets of subgroups

G _{1,\ldots,G} _{k} of G .

Herzog and Schönheim conjecturedthat if

A forms a partition of G with k> 1 ,then the (finite) indices [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 H is any subgroup of G with index k = [G:H]< inf ty then G can be partitioned into k left cosets of H .

Subnormal subgroups

In 2004, Zhi-Wei Sun proved an special caseof the Herzog–Schönheim conjecture in the case where

G _{1,\ldots,G} _{k} are subnormal in G .^[2] A basic lemma in Sun's proof states that if G _{1,\ldots,G} _{k} are subnormal and of finite index in G , then[ G: c ap ^{kG} _{i ] | \prod} ^{k[G:G} ^{i]}

and hence

P ( [ G: c ap ^{kG} _{i ] ) = c up} ^{kP([G:G} ^{i]),}

where

P(n) denotes the set of primedivisors of n .

Mirsky–Newman theorem

When

G is the additive group \Z of integers, the cosets of 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]

Notes and References

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