# 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^{ -1}gf and g^{ -1}fg. 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.

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