The parity of a permutation records whether it is a product of an even or odd number of transpositions. It is well-defined and equals the sign of the corresponding permutation matrix.
The sign homomorphism sends an even permutation to and an odd permutation to ; its kernel is the alternating group.
A symmetric group is the group of all bijections from a set to itself, with function composition as its operation.
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The parity of a permutation refers to whether the permutation is even or odd based on the number of transpositions it can be decomposed into. - A **transposition** is a permutation that swaps two elements while leaving all others unchanged. - A permutation is classified as **even** if it can be expressed as a product of an even number of transpositions, and it is classified as **odd** if it can be expressed as a product of an odd number of transpositions.
