Editing Anticommutative property (section)

== Properties ==
If the abelian group <math>B</math> has no 2-[[Torsion (algebra)|torsion]], implying that if <math>x = -x</math> then <math>x = 0</math>, then any anticommutative bilinear map <math>f\colon A^2 \to B</math> satisfies

:<math>f(x, x) = 0.</math>

More generally, by [[Transposition (mathematics)|transposing]] two elements, any anticommutative multilinear map <math>g\colon A^n \to B</math> satisfies

:<math>g(x_1, x_2, \dots x_n) = 0</math>

if any of the <math>x_i</math> are equal; such a map is said to be '''[[Alternating multilinear map|alternating]]'''. Conversely, using multilinearity, any alternating map is anticommutative. In the binary case this works as follows: if <math>f\colon A^2 \to B</math> is alternating then by bilinearity we have

:<math>f(x+y, x+y) = f(x, x) + f(x, y) + f(y, x) + f(y, y) = f(x, y) + f(y, x) = 0</math>

and the proof in the multilinear case is the same but in only two of the inputs.