Transposition (mathematics)

Given a finite set X={a_1,a_2,ldots,a_n}, a transposition is a permutation (bijective function of X onto itself) f, such that there exist indices i, j such that f(a_i) = a_j, f(a_j) = a_i and f(a_k) = a_k for all other indices k. This is often denoted (in the cycle notation) as (a, b).

Related Topics:
Finite - Set - Permutation - Bijective function - Onto - Cycle notation

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Example: If X={a, b, c, d, e} the function sigma given by

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:egin{matrix} sigma(a)&=&a\ sigma(b)&=&e\ sigma(c)&=&c\ sigma(d)&=&d\ sigma(e)&=&b end{matrix}

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is a transposition.

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One of the main results on symmetric groups states that any permutation can be expressed as the composition (product) of transpositions, and for any two decompositions of a given permutation, the number of transpositions is always even or always odd.

Related Topics:
Symmetric group - Composition - Decompositions

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