Herzog–Schönheim conjecture

Herzog–Schönheim conjecture   (Redirected from Mirsky–Newman theorem) Jump to navigation Jump to search 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]

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