Group multiplication of permutations

The applet below is a device to help master the group multiplication of permutations; it serves as an illustration to Lemma 1. The applet randomly generates pairs of permutations f and g and displays their products fg and gf as well as their conjugates f -1gf and g -1fg. Do not forget that, in a product, permutations are executed left to right.

The applet allows for two representations of permutations: as an ordered list of values {f(1), ..., f(n)} or as a product of cycles. The latter is particularly useful as an illustration of Lemma 1. However note that a cycle admits various representations that are obtained from each other by rotating its list of elements. For example, the cycles (1 3 8 4), (3 8 4 1), (8 4 1 3), and (4 1 3 8) represent the same cycle. Cycles generated by the applet always start with the smallest element.

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at, download and install Java VM and enjoy the applet.

What if applet does not run?


  • Transpositions
  • Groups of Permutations
  • Sliders
  • Puzzles on graphs
  • Equation in Permutations

    Related material

  • What Is Multiplication?
  • Addition and Multiplication Tables in Various Bases
  • Peasant Multiplication
  • Long Multiplication - an Interactive Gizmo
  • Egyptian Multiplication
  • Lattice Multiplication - an Interactive Gizmo
  • Using Math Rules: An Example
  • Tables for Multiplication
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