Hierarchical Complexity of Finite Groups

TL;DR

This paper introduces a hierarchical complexity measure for finite groups based on their atomic simple group constituents, providing a rigorous, computable framework to understand their structure and complexity bounds.

Contribution

It axiomatizes hierarchical complexity for finite groups, proves the existence of a unique maximal complexity function, and relates it to minimal decompositions and existing group invariants.

Findings

The maximal complexity function is unique and effectively computable.

Hierarchical complexity bounds and relates to socle length and other invariants.

Minimal decompositions are not necessarily unique in components or order.

Abstract

What are simplest ways to construct a finite group from its atomic constituents? To understand part-whole relations between finite simple groups and the global structure of finite groups, we axiomatize complexity measures on finite groups. From the Jordan-H\"older 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-H\"older factors, each with well-defined multiplicities through iterated extension. 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 answer this question rigorously, we give a natural set of hierarchical complexity axioms for finite groups, and prove these axioms are satisfied by a unique maximal complexity function $\mathbf{cx}$. 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. This hierarchical complexity is thus effectively computable, and bounded below by all other complexity measures satisfying the axioms, including generalizations of derived length and Fitting height. For solvable groups, the unique maximal group complexity measure satisfying the axioms agrees with the restriction of the one for all finite groups, and in addition satisfies an embedding axiom. 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, with a canonical decomposition. 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.