Hierarchical Complexity of Finite Groups
Chrystopher L. Nehaniv

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 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 . We prove this function is the same as…
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Taxonomy
TopicsFinite Group Theory Research · Geometric and Algebraic Topology · graph theory and CDMA systems
