We prove new upper bounds for the length of laws that hold for all groups of size at most $n$ -- improving on previous results of Bou-Rabee and Kassabov-Matucci. The methods make use of the classification of finite simple groups. Stronger bounds are proved in case the groups are assumed to be nilpotent or solvable.
On the number of classes of conjugate Hall subgroups in finite simple groups
Commutativity degree, its generalizations, and classification of finite groups
Towards the Classification of Finite Simple Groups with exactly Three or Four Supercharacter Theories
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