High Quality Content by WIKIPEDIA articles! High Quality Content by WIKIPEDIA articles! In mathematics, Zolotarev's lemma in number theory states that the Legendre symbolfor an integer a modulo a prime number p, can be computed aswhere ? denotes the signature of a permutation and ?a the permutation of the residue classes mod p induced by modular multiplication by a, provided p does not divide a.In general, for any finite group G of order n, it is easy to determine the signature of the permutation ?g made by left-multiplication by the element g of G. The permutation ?g will be even, unless there are an odd number of orbits of even size. Assuming n even, therefore, the condition for ?g to be an odd permutation, when g has order k, is that n/k should be odd, or that the subgroup <g> generated by g should have odd index.

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