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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,

$\displaystyle (abc)$ $\textstyle =$ $\displaystyle (ab)(ac)$  
$\displaystyle (abcde)$ $\textstyle =$ $\displaystyle (ab)(ac)(ad)(ae).$  

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

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

© 1996-9 Eric W. Weisstein