Group theory 30: Outer automorphisms

Group theory 30: Outer automorphisms

by Richard E Borcherds

Formal & Physical Sciences Mathematics PBMathematicsPBGGroups and group theory
🎙 Richard E Borcherds 👥 82K 📅 July 2, 2020 ⏱ 24 min 👁 4K 📄 lecture 🧭 2026-08-17
Available in: English (current) Français

Keywords

outer automorphismsymmetric groupS6inner automorphismautomorphism group

Summary

This lecture by Richard Borcherds explores outer automorphisms of symmetric groups, focusing on the exceptional case of S6. It begins by defining inner and outer automorphisms, and the exact sequence relating them. The lecturer then discusses ‘obvious’ automorphisms for groups like PSL(n,q) and alternating groups, noting that for most n, Aut(A_n) is S_n, except for A6 which has extra outer automorphisms. The main goal is to show that S6 has an outer automorphism of order 2, while other symmetric groups do not. The proof involves constructing a transitive subgroup of S6 isomorphic to S5, leading to a homomorphism S6 to S6 that is not inner. An explicit outer automorphism is given using generators and relations, mapping transpositions to products of three transpositions. The lecture concludes by showing that for n>6, the number of transpositions differs from other conjugacy classes, preventing outer automorphisms, and that the outer automorphism group of S6 has order 2.

153 words

Critical Evaluation

Value of the Information & Strength of the Argument

The lecture provides a thorough and rigorous treatment of outer automorphisms, with clear definitions and detailed proofs. The argumentation is solid, building from basic concepts to the exceptional case of S6. The explicit construction of the outer automorphism is particularly valuable, as it makes the abstract concept tangible. The use of conjugacy class sizes to rule out outer automorphisms for other symmetric groups is elegant and convincing.

Scientific Rigor, Source Quality, Title Accuracy

The lecture is mathematically rigorous, with all statements justified. The sources are primarily the lecturer’s own knowledge and standard group theory results, though no external references are cited. The title accurately reflects the content, which is focused on outer automorphisms of symmetric groups. The lecture is well-structured and accessible to those with a background in group theory.

139 words

Title / Content Match

The title accurately reflects the content, focusing on outer automorphisms of symmetric groups.

Quality & Reliability

9/10

Lecture by a renowned mathematician, rigorous and clear, with detailed proofs and examples.

Key Moments

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

Contribution & Novelties

This lecture provides a clear and detailed exposition of a classic result in group theory: the existence of outer automorphisms for S6. The explicit construction of the outer automorphism is particularly illuminating, as it gives a concrete example of a non-inner automorphism. The lecture also highlights the connection to sporadic groups like M11.

Pour aller plus loin :

  • Outer automorphism group — Wikipedia article providing background on outer automorphisms.
  • Symmetric group — Wikipedia article on symmetric groups, including properties and automorphisms.
  • Mathieu group M11 — Wikipedia article on the Mathieu group M11, which relates to outer automorphisms of A6.

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Radar Profile

The radar profile shows very high scores across all dimensions, indicating a lecture that is both informative and rigorous, with a strong technical level and high reliability.

Reliability 10/10