Group theory 26: Too many p groups

Group theory 26: Too many p groups

by Richard E Borcherds

Formal & Physical Sciences Mathematics PBMathematicsPBGGroups and group theory
🎙 Richard E Borcherds 👥 82K 📅 July 2, 2020 ⏱ 12 min 👁 3K 📄 lecture 🧭 2026-08-17
Available in: English (current) Français

Keywords

p-groupsclassificationgroup theoryenumerationcommutators

Summary

In this lecture, Richard Borcherds explains why p-groups are not classified in general, focusing on the enormous number of groups of order p^n. He begins by noting that groups of order 32 are already numerous (51) and that a comprehensive classification exists in a large reference book by Marshall Hall and James Senior. He then shows that for higher powers, the number of groups grows extremely fast, making classification impractical. He constructs a lower bound for the number of groups of order p^n by considering extensions of elementary abelian groups. He estimates the number of possible commutator maps, which gives a rough count of p^(2/27 n^3) groups, after accounting for symmetries. He concludes that for large n, the number of groups is so vast that classification is not feasible, and he mentions that future lectures will focus on more interesting orders like 48 and 60.

145 words

Critical Evaluation

Value of the Information & Strength of the Argument

The lecture provides valuable insight into why p-groups are not classified, using a clear and rigorous argument. The reasoning is well-structured, starting from concrete examples and building to a general estimate. The derivation of the lower bound is logical, and the handling of symmetries is appropriately discussed. The argument is convincing and highlights the impracticality of classification for large n.

Scientific Rigor, Source Quality, Title Accuracy

The lecture is scientifically rigorous, with a solid mathematical foundation. The main source mentioned is the book by Marshall Hall and James Senior, which is a standard reference. The title accurately reflects the content. No external sources are cited, but the mathematical reasoning is self-contained and reliable.

122 words

Title / Content Match

The title accurately reflects the content, which explains why p-groups are too numerous to classify.

Quality & Reliability

8/10

The lecture is part of a formal mathematics course by a renowned mathematician. The content is rigorous and based on established group theory. The argument is well-structured and the estimates are clearly derived. The main limitation is the lack of formal citations, but the mathematical reasoning is sound.

Key Moments

Markers derived by PSI from the transcript: the creator did not define chapters.

Cited Sources

  • The Groups of Order 2^n (n ≤ 6) — Book by Marshall Hall and James Senior, mentioned as a reference for groups of order up to 64.

Concurring Sources

  • Higman's PORC conjecture — Related to the growth of the number of p-groups.

Contribution & Novelties

This lecture provides a clear and accessible explanation of why p-groups are not classified, using a constructive lower bound. The original contribution is the pedagogical presentation of the counting argument, which is often scattered in the literature. It bridges the gap between abstract theory and practical considerations.

Pour aller plus loin :

  • Classification of finite simple groups — Relevant background on classification efforts in group theory.
  • p-group — Basic definitions and properties of p-groups.
  • Higman’s PORC conjecture — Related to the number of p-groups of a given order.

88 words

Radar Profile

The radar profile shows high scores in quality and technical level, with moderate quantity of information and reliability. This indicates a focused, rigorous lecture that may not cover a wide range of topics but provides deep insight.

Reliability 8/10

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