$\newcommand{\aatavdots}[1]{\vdots} \newcommand{\codeword}{} \newcommand{\identity}{\mathrm{id}} \newcommand{\notdivide}{\nmid} \newcommand{\notsubset}{\not\subset} \newcommand{\lcm}{\operatorname{lcm}} \newcommand{\gf}{\operatorname{GF}} \newcommand{\inn}{\operatorname{Inn}} \newcommand{\aut}{\operatorname{Aut}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\cis}{\operatorname{cis}} \newcommand{\chr}{\operatorname{char}} \newcommand{\Null}{\operatorname{Null}} \newcommand{\transpose}{\text{t}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \definecolor{fillinmathshade}{gray}{0.9} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} $

Chapter 9 The Simplicity of $A_n$

A group with no nontrivial proper normal subgroups is simple. The cyclic groups ${\mathbb Z}_p\text{,}$ $p$ prime, are trivially simple, since they have no proper subgroups at all. Our goal in this chapter is a much less trivial example: the alternating group $A_n$ is simple for $n \geq 5\text{.}$ Along the way we develop the tools needed to state this precisely — cycle notation, the parity of a permutation, and the alternating group itself.

This chapter was assembled by Claude, Anthropic’s AI model, at the course instructor’s direction, combining the simplicity proof with the permutation-theory prerequisites it needs, adapted from Thomas W. Judson’s original text.

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Chapter 9 The Simplicity of \(A_n\)