Actions and Sylow theorems · Definition

Cycle of a permutation

modern algebragroups

Before you start you need to know

Cycles are the building blocks of every permutation and the most compact way to write one.

Definition

A cycle of length \(k\) in \(S_n\) is a permutation sending \(a_1\mapsto a_2\mapsto\cdots\mapsto a_k\mapsto a_1\) and fixing all other elements.

\[(a_1\ a_2\ \cdots\ a_k)\]

The cycle \((a_1\ a_2\ \cdots\ a_k)\) does not change if we start from another of its elements: \((1\ 2\ 3)=(2\ 3\ 1)=(3\ 1\ 2)\). A cycle of length 2 is called a transposition, and one of length 1 is the identity. The inverse is obtained by running the cycle backwards, \((a_1\ a_2\ \cdots\ a_k)^{-1}=(a_k\ \cdots\ a_2\ a_1)\).

Two cycles are disjoint if they share no elements, and then they commute. If they share elements, the order matters: composing from right to left, \((1\ 2)(2\ 3)=(1\ 2\ 3)\), whereas \((2\ 3)(1\ 2)=(1\ 3\ 2)\).

Every permutation of a finite set is a product of disjoint cycles, uniquely up to the order of the factors. To find it, start from an element, follow its path until it returns, and repeat with an element not yet visited; fixed elements are not written. For example, the list \([2,1,4,5,3,6]\) is \((1\ 2)(3\ 4\ 5)\).

Think about it

How many cycles of length \(k\) are there in \(S_n\)?

There are \(\binom{n}{k}(k-1)!\): choose the \(k\) elements and then arrange them in a circle, ignoring the starting point.

Exercises 10

  1. In \(S_5\), the cycle \((1\ 3\ 5)\) sends \(3\) to…
  2. What is the length of the cycle \((2\ 5\ 3\ 1)\)?
  3. Which of these cycles is the same as \((1\ 2\ 3\ 4)\)?
  4. What is the inverse of the cycle \((1\ 2\ 3\ 4)\)?
  5. In \(S_3\), what is \((1\ 2)(2\ 3)\) (the right-hand cycle is applied first)?
  6. Which statement about cycles in \(S_n\) is true?
  7. How many distinct cycles of length \(3\) are there in \(S_4\)?
  8. In \(S_6\), the permutation with list \([2,1,4,5,3,6]\) is written as a product of disjoint cycles as:
  9. In \(S_4\), the square of the cycle \((1\ 2\ 3\ 4)\) (the cycle applied twice) is…
  10. How many elements of \(\{1,\dots,6\}\) does the cycle \((1\ 4\ 2)\) in \(S_6\) fix?

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