• Corpus ID: 237372485

Hierarchical Complexity of Finite Groups

@inproceedings{Nehaniv2021HierarchicalCO,
  title={Hierarchical Complexity of Finite Groups},
  author={Chrystopher L. Nehaniv},
  year={2021}
}

• Chrystopher L. Nehaniv
• Published 26 August 2021
• Mathematics

What are simplest ways to construct a finite group from its atomic constituents? To understand part-whole relations between finite simple groups (“atoms”) and the global structure of finite groups, we axiomatize complexity measures on finite groups. From the Jordan-Hölder theorem and Frobenius-Lagrange embedding in an iterated wreath product, any finite group G can be constructed from a unique collection of simple groups, its Jordan-Hölder factors, each with well-defined multiplicities through iterated extension by simple groups. What is the least number of levels needed in such a hierarchical construction if a level is allowed to include several of these atomic pieces? To pose and answer this question rigorously, we give a natural set of hierarchical complexity axioms for finite groups relating to constructability, extension, quotients, and products, and prove these axioms are satisfied by a unique maximal complexity function cx. We prove this function is the same as the minimal number of “spans of gems” (direct products of simple groups) in a subnormal series with all factors of this type. Hierarchical complexity cx is thus effectively computable, and bounded below by all other complexity measures satisfying the axioms, including generalizations of derived length, Fitting height and solvability. Also, the hierarchical complexity of a normal subgroup is bounded above by the complexity of the whole group, although this is not assumed in the axioms and does not follow from them. For solvable groups, the unique maximal group complexity measure defined satisfying the axioms agrees with the restriction of the one for all finite groups, and in addition satisfies an embedding axiom which decidedly cannot be applied in the general case of all finite groups. In both cases, the complexity of a group is bounded above and below by various natural functions. In particular, hierarchical complexity is sharply bounded above by socle length, which yields a canonical decomposition and satisfies all the axioms except the extension axiom. Examples illustrate applications of the bounds and axiomatic methods in determining complexity of groups. We show also that minimal decompositions need not be unique in terms of what components occur nor their ordering. The complexity axioms are also shown to be independent. This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), funding reference number RGPIN-2019-04669. Cette recherche a été financée par le Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG), numéro de référence RGPIN-2019-04669. The author also is grateful to Mr. Thomas George, recipient of a University of Waterloo President’s Award as undergraduate research assistant, for assistance with constructing software tools with which the counterxample groups in this paper were initially found and studied by the author.