The Timpson Lyceum · Discrete Math (2025)

Permutations: Example of a Group - Lecture 13 (Discrete Math)

Overview

A permutation of a set is simply a bijection from the set to itself — a reordering of its elements. This lecture introduces permutations as the central example of an algebraic group. After a quick review of functions, one-to-one and onto properties, and function composition, we collect all permutations of $\{1,2,\dots,n\}$ into the set $S_n$ and verify the four group axioms (closure, associativity, identity, inverses). We then develop efficient ways to write permutations — two-row notation and the favored cyclic notation — and study transpositions (single swaps). The deepest result is that the number of transpositions needed to build a given permutation always has a fixed parity (always even or always odd), which lets us define the sign of a permutation and read it straight off the cycle structure.

Key Concepts

Function review. A function $f:A\to B$ assigns each input in $A$ exactly one output in $B$ (equal inputs force equal outputs). The domain is all of $A$; the image is a subset of $B$.

• One-to-one (injective): equal outputs force equal inputs — $f(a)=f(b)\Rightarrow a=b$.
• Onto (surjective): the image is all of the codomain $B$.
• Bijection: both one-to-one and onto.
• Composition: $(g\circ f)(a)=g\big(f(a)\big)$ — do $f$ first, then $g$ (read right to left).

Permutation. For a set $A$, a permutation on $A$ is a function $\pi:A\to A$ that is a bijection. Equivalently, it is a reordering of the elements of $A$. Permutation variables use lowercase Greek letters: $\pi,\sigma,\tau,\iota$.

Counting permutations. A set of $n$ elements has exactly $n!$ permutations (choose the first image in $n$ ways, the next in $n-1$, …, down to $1$).

The set $S_n$. $S_n$ denotes the set of all permutations of $\{1,2,\dots,n\}$. So $|S_n| = n!$. The identity permutation $\iota$ maps every element to itself; $\iota$ is always a bijection, hence always lies in $S_n$.

$S_n$ is a group. $S_n$ together with composition $\circ$ satisfies the group axioms:

• Closure: $\pi,\sigma\in S_n \Rightarrow \sigma\circ\pi\in S_n$ (a bijection composed with a bijection is a bijection).
• Associativity: $\tau\circ(\sigma\circ\pi)=(\tau\circ\sigma)\circ\pi$.
• Identity: $\iota\circ\pi=\pi=\pi\circ\iota$.
• Inverses: every $\pi$ has an inverse $\pi^{-1}\in S_n$ with $\pi\circ\pi^{-1}=\iota=\pi^{-1}\circ\pi$.

Notation for a permutation.

• Two-row: top row $1\,2\,3\,4\,5$, bottom row the images. Since the top is always $1\dots n$, the bottom row alone determines $\pi$.
• Cyclic notation: start with the smallest unused element, follow where it maps until the cycle closes, open a new parenthesis for the next unused element. Example: $\pi=(1\;2\;4\;3)(5)$ means $1\to2\to4\to3\to1$ and $5\to5$.
• Inverse from cycles: reverse each cycle. The inverse of $(1\;2\;7)(3\;5)(4\;8\;6)(9)$ is $(7\;2\;1)(5\;3)(6\;8\;4)(9)$.

Transposition. A permutation $\tau\in S_n$ is a transposition (a single swap) if there exist distinct $i,j$ with $\tau(i)=j$, $\tau(j)=i$, and $\tau(k)=k$ for every other $k$. In cyclic notation a transposition is written as its one nontrivial cycle, e.g. $(2\;4)$.

Decomposition into transpositions. Every permutation can be written as a composition of transpositions (the "swaps" that turn the identity ordering into the target ordering). This is like factoring a number into primes — but the factorization is not unique, and you can always pad it (e.g. $(1\;3)(1\;3)=\iota$ does nothing), so a permutation can be expressed using infinitely many different counts of transpositions.

Even / odd / sign. A permutation is even if it can be written as an even number of transpositions, odd if as an odd number. This parity is well-defined: if $\pi$ can be written with $a$ transpositions one way and $b$ another, then $a\equiv b \pmod 2$. The sign is

• Sign is multiplicative: $\operatorname{sgn}(\sigma\circ\pi)=\operatorname{sgn}(\sigma)\,\operatorname{sgn}(\pi)$ (because $(-1)^{x+y}=(-1)^x(-1)^y$).
• Parity formula: if $c(\pi)$ is the number of cycles of $\pi$ (each fixed point counts as a cycle), then the number of transpositions $a$ satisfies $a \equiv n - c(\pi) \pmod 2$. So parity is readable directly from the cycle structure.

Cycle-counting lemma. Composing a permutation with one transposition changes its cycle count by exactly $\pm 1$: it glues two cycles into one (if the swapped pair lies in different cycles) or splits one cycle into two (if they lie in the same cycle). This is the engine behind the parity proof.

Worked Example

Decompose a permutation into transpositions and find its sign. Take $\pi=(1\;2\;3\;4\;5)\in S_5$ (the 5-cycle $1\to2\to3\to4\to5\to1$). Build it as a chain of swaps, then apply the parity formula.

The chain reorders $1\,2\,3\,4\,5$ into $2\,3\,4\,5\,1$ exactly as $\pi$ does; reading $3$ right-to-left through the swaps lands on $4$, confirming $\pi(3)=4$. Both the count of transpositions ($4$) and the formula $n-c(\pi)=4$ give the same even parity, so $\operatorname{sgn}(\pi)=+1$.

Summary

• A permutation of a set is a bijection from the set to itself — equivalently, a reordering of its elements; an $n$-element set has $n!$ of them.
• $S_n$, the set of all permutations of $\{1,\dots,n\}$ under composition, is a group: closed, associative, with identity $\iota$ and an inverse for every element. It is the standard concrete example of a group.
• Cyclic notation is the compact way to record a permutation; its inverse is obtained by reversing each cycle.
• A transposition is a single swap; every permutation factors into transpositions, though not uniquely and not with a unique count.
• The parity (even/odd) of a permutation is well-defined, giving the sign $\operatorname{sgn}(\pi)=(-1)^a$, which is multiplicative over composition.
• Parity can be read instantly from cycle structure via $a \equiv n - c(\pi)\pmod 2$, since each transposition changes the number of cycles by exactly one.