From Farey symbols to generators for subgroups of finite index in integral group rings of finite groups

DOI10.1017/is009012013jkt079zbMath1209.16030OpenAlexW2035628391MaRDI QIDQ3061169

Ann Dooms, Eric Jespers, Olexandr Konovalov

Publication date: 14 December 2010

Published in: Journal of K-Theory (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1017/is009012013jkt079

zbMATH Keywords

finite groupsgeneratorssubgroups of finite indexfree subgroupsgroups of unitsunit groupsintegral group ringscongruence subgroupsrational group algebrassimple Wedderburn componentsFarey symbolsFarey units

Mathematics Subject Classification ID

Group rings (16S34) Unimodular groups, congruence subgroups (group-theoretic aspects) (20H05) Group rings of finite groups and their modules (group-theoretic aspects) (20C05) Structure of modular groups and generalizations; arithmetic groups (11F06) Software, source code, etc. for problems pertaining to group theory (20-04) Computational aspects of associative rings (general theory) (16Z05) Units, groups of units (associative rings and algebras) (16U60)

Related Items

Concentration of closed geodesics in the homology of modular curves, From the Poincaré Theorem to generators of the unit group of integral group rings of finite groups

Uses Software

• GAP
• wedderga
• Congruence

Cites Work

• Unnamed Item
• Arithmetic subgroups of algebraic groups
• A theorem on units of integral group rings
• Classification of finite subgroups of \(2\times 2\) matrices over a division algebra of characteristic zero
• Construction of units in group rings of monomial and symmetric groups
• Generators of large subgroups of the unit group of integral group rings
• Units of group rings of solvable and Frobenius groups over large rings of cyclotomic integers
• Embedding free products in the unit group of an integral group ring.
• Degree 1 and 2 representations of nilpotent groups and applications to units of group rings
• Linear groups and group rings.
• Constructing free subgroups of integral group ring units
• Describing units of integral group rings of some 2-groups
• Construction of Units in Integral Group Rings of Finite Nilpotent Groups
• UNITARY UNITS IN INTEGRAL GROUP RINGS
• Free Subgroups in the Unit Groups of Integral Group Rings
• An Arithmetic-Geometric Method in the Study of the Subgroups of the Modular Group
• Construction of units in integral group rings of finite nilpotent groups
• On free subgroups of units of rings
• Groups generated by two bicyclic units in integral group rings
• An Algorithm for Determining if a Subgroup of the Modular Group is Congruence
• Central units of integral group rings of nilpotent groups
• SPECIAL POLYGONS FOR SUBGROUPS OF THE MODULAR GROUP AND APPLICATIONS
• Free Groups and Subgroups of Finite Index in the Unit Group of an Integral Group Ring
• Bicyclic units, Bass cyclic units and free groups