Actions and Sylow theorems · Definition

Symmetric group

modern algebragroups

Before you start you need to know

It is the universal example of a group: every finite group lives inside one of them, and its elements are the rearrangements of a list.

Definition

The symmetric group \(S_n\) is the set of bijections \(\sigma\) of \(\{1,\dots,n\}\) onto itself, with composition \(\sigma\tau\) as the operation.

\[(\sigma\tau)(i)=\sigma(\tau(i))\]

A permutation \(\sigma\) can be described by listing its images \([\sigma(1),\dots,\sigma(n)]\). The identity is neutral, the inverse of \(\sigma\) sends each image back to its source, and associativity is inherited from composition of functions. Choosing the list amounts to choosing a distinct image for each element, so \(|S_n|=n!\).

Here we compose from right to left: in \(\sigma\tau\) the map \(\tau\) is applied first. This convention matters because, from \(n=3\) on, the operation stops being commutative: \(S_3\) has six elements and is the smallest nonabelian group. The groups \(S_1\) and \(S_2\) are abelian.

Cayley's theorem says every finite group of order \(n\) is isomorphic to a subgroup of \(S_n\), so studying symmetric groups is, in a sense, studying all finite groups. Next we will decompose their elements into cycles and assign each a sign.

Think about it

How many elements of \(S_4\) send \(1\) to \(2\)?

With that image fixed, the remaining three elements are permuted freely: \(3!=6\).

Exercises 10

  1. How many elements does \(S_4\) have?
  2. Write a permutation of \(S_3\) as the list \([\sigma(1),\sigma(2),\sigma(3)]\). If \(\sigma=[2,3,1]\) and \(\tau=[2,1,3]\), what is \(\sigma\tau\) (\(\tau\) is applied first)?
  3. What is the identity element of \(S_5\)?
  4. If \(\sigma=[2,3,1,4]\) in \(S_4\) (list \([\sigma(1),\ldots,\sigma(4)]\)), what is \(\sigma^{-1}\)?
  5. What is the smallest \(n\) for which \(S_n\) is not abelian?
  6. How many permutations \(\sigma\in S_5\) satisfy \(\sigma(1)=1\)?
  7. Which statement about \(S_n\) is true?
  8. For which \(n\) does \(S_n\) have exactly \(120\) elements?
  9. In \(S_4\) let \(\sigma=[2,1,3,4]\) and \(\tau=[1,3,2,4]\). What is \((\sigma\tau)(2)\)?
  10. In \(S_4\) with \(\sigma=[2,3,4,1]\) and \(\tau=[1,2,4,3]\), which permutation \(x\) satisfies \(\sigma x=\tau\)?

References

  • David S. Dummit, Richard M. Foote, Abstract Algebra, 3rd edition, Wiley, 2004. §1.3
  • Joseph A. Gallian, Contemporary Abstract Algebra, 9th edition, Cengage, 2017. cap. 5
  • I. N. Herstein, Abstract Algebra, 3rd edition, Wiley, 1996. §2.10
  • John B. Fraleigh, A First Course in Abstract Algebra, 7th edition, Pearson, 2002. §8