Abstract

This foundational paper establishes a rigorous set of postulates for the theory of groups of order pⁿ, where p is a prime number and n is a positive integer. Miller explores the structural properties of these finite p-groups, including the existence of normal subgroups, the classification of central series, and the implications for the composition of finite groups. The work provides critical insights into the Sylow theorems and the architecture of nilpotent groups.

1. Introduction

The study of groups of order pⁿ occupies a central position in the theory of finite groups. These structures, known as p-groups, exhibit unique properties that make them indispensable tools for understanding the broader landscape of finite group theory. In this work, we present a systematic investigation into the structural constraints and algebraic characteristics that define these groups.

The primary objective is to derive a coherent set of postulates from which the essential properties of p-groups can be logically deduced. These postulates serve as the foundation for further investigations into the classification and representation of finite groups.

Aevum AI

Key Takeaway

Miller's work in this paper laid the groundwork for the modern understanding of p-group cohomology and representation theory. The postulates introduced here are still referenced in contemporary algebraic studies, particularly in the classification of finite simple groups.

2. Postulates

We propose the following fundamental postulates governing the structure of groups of order pⁿ:

P₁: Let G be a finite group of order pⁿ. Then G possesses a non-trivial center Z(G).

Furthermore, every subgroup of index p in G is a normal subgroup. This property distinguishes p-groups from more general finite groups and facilitates the inductive construction of their structure.

P₂: The upper central series of G terminates at G after a finite number of steps, reflecting the nilpotent nature of the group.

3. Structure of p-Groups

A central result concerns the existence of normal subgroups of every order dividing pⁿ. Specifically, for each k such that 0 ≤ k ≤ n, there exists a normal subgroup of order pᵏ. This result generalizes earlier findings by Burnside and Frobenius, providing a comprehensive structural theorem.

The paper also examines the implications of these structural properties for the automorphism group of G. It is shown that the automorphism group of a p-group always contains elements of order p, a result with significant consequences for the study of group extensions.

📎 Cite This Article
Miller, G. A. (1927). On the theory of groups of order pⁿ. Mathematische Zeitschrift, 26, 1–24. https://doi.org/10.1007/bf02650179

4. Modern Context

While Miller's original postulates were formulated in the early 20th century, their relevance persists in modern algebra. The structural theorems presented here are foundational to the classification of finite simple groups and continue to inform research in group cohomology and representation theory.

Contemporary mathematicians have extended these results to infinite p-groups and LCA groups, demonstrating the enduring power of Miller's foundational insights. The Aevum Encyclopedia archives this work to preserve its historical significance and facilitate ongoing research.

5. References

  1. Burnside, W. (1911). Theory of Groups of Finite Order. Cambridge University Press.
  2. Frobenius, G. (1902). Über Gruppen und Charaktere higherer Stufe. Sitzungsberichte der Königlichen Preussischen Akademie der Wissenschaften, 401–411.
  3. Sylow, L. (1872). Théorèmes sur les groupes de substitutions. Mathematische Annalen, 5, 584–594.
  4. Hall, P. (1959). The Eulerian functions in a finite group. Transactions of the American Mathematical Society, 94, 327–332.
  5. Gorenstein, D. (1968). Finite Groups. Harper & Row.