Permutation Group

A finite Group of substitutions of elements for each other. For instance, the order 4 permutation group $\{4, 2, 1, 3\}$ would rearrange the elements $\{A, B, C, D\}$ in the order $\{D, B, A, C\}$. A Substitution Group of two elements is called a Transposition. Every Substitution Group with $>2$ elements can be written as a product of transpositions. For example,

$$(abc) = (ab)(ac)$$

$$(abcde) = (ab)(ac)(ad)(ae).$$

Conjugacy Classes of elements which are interchanged are called Cycles (in the above example, the Cycles are $\{\{1, 3, 4\}, \{2\}\}$).

See also Cayley's Group Theorem, Cycle (Permutation), Group, Substitution Group, Transposition