A string group generated by involutions, or SGGI, is a pair τ D .G; S /, where G is a group and S D 1Ƿ0;:::; Ƿr -1ºis an ordered set of involutions generating G and satisfying the following commuting property: for all i; j 2 10;:::; r - 1º, ji - j j ¤ 1 implies .Ƿi Ƿj /2 D 1. When S is an independent set, the rank of τ is the cardinality of S . We determine an upper bound for the rank of an SGGI over the alternating group of degree n. Our bound is tight when n ＝ 0; 1; 4 .mod 5/.
Anzanello, J., Fernandes, M., Spiga, P. (2026). The maximal rank of a string group generated by involutions for alternating groups. JOURNAL OF GROUP THEORY [10.1515/jgth-2025-0080].
The maximal rank of a string group generated by involutions for alternating groups
Anzanello J.;Spiga P.
2026
Abstract
A string group generated by involutions, or SGGI, is a pair τ D .G; S /, where G is a group and S D 1Ƿ0;:::; Ƿr -1ºis an ordered set of involutions generating G and satisfying the following commuting property: for all i; j 2 10;:::; r - 1º, ji - j j ¤ 1 implies .Ƿi Ƿj /2 D 1. When S is an independent set, the rank of τ is the cardinality of S . We determine an upper bound for the rank of an SGGI over the alternating group of degree n. Our bound is tight when n ＝ 0; 1; 4 .mod 5/.
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