Anticommutative property
In mathematics, anticommutativity is a specific property of some non-commutative operations. In mathematical physics, where symmetry is of central importance, these operations are mostly called antisymmetric operations, and are extended in an associative setting to cover more than two arguments. Swapping the position of two arguments of an antisymmetric operation yields a result which is the "inverse" of the result with unswapped arguments. The notion "inverse" refers to a group structure on the operation's codomain, possibly with another operation, such as addition. Subtraction is an anticommutative operation because −("a" − "b") = "b" − "a". For  example, 2 − 10 = −(10 − 2) = −8. A prominent example of an anticommutative operation is the Lie bracket. Definition. If formula_1 are two abelian groups, a bilinear map formula_2 is anticommutative if for all formula_3 we have More generally, a multilinear map formula_5 is anticommutative if for all formula_6 we have where formula_8 is the sign of the permutation formula_9. Properties. If the abelian group formula_10 has no 2-torsion, implying that if formula_11 then formula_12, then any anticommutative bilinear map formula_2 satisfies More generally, by transposing two elements, any anticommutative multilinear map formula_5 satisfies if any  of the formula_17 are equal; such a map is said to be alternating. Conversely, using multilinearity, any alternating map is anticommutative. In the binary case this works as follows: if formula_2 is alternating then by bilinearity we have and the proof in the multilinear case is the same but in only two of the inputs. Examples. Examples of anticommutative binary operations include: