# Composition of Permutations

One of the reasons why the symmetric group Sym(n) is so interesting to group theorists is that it is relatively easy to understand, but non-abelian. This applet allows students to experiment with composing two permutations in Sym(4) to explore the properties of this group.

Students select two permutations, one from each drop-box. The composition of these permutations is shown both in cycle notation and by following the arrows from the left side all the way to the right.

Using the applet, the students can explore many concepts:

• How do we compute the composition of two non-disjoint cycles? How can we determine, for example, that (1 2 3)(2 3 4) = (1 3)(2 4)?
• Is Sym(4) abelian?
• Why is the alternating group Alt(4) a subgroup of Sym(4)? That is, can we demonstrate that the composition of two even permutations is always even?
• Why is the set of odd permutations not a subgroup of Sym(4)? Can we show that the composition of two odd permutations is not always odd?

This page was created by James Hamblin, Associate Professor of Mathematics at Shippensburg University.