general commutativity

Theorem.  If the binary operationMathworldPlanetmath” on the set S is commutativePlanetmathPlanetmathPlanetmath, then for each  a1,a2,,an in S and for each permutationMathworldPlanetmath π on  {1, 2,,n},  one has

i=1naπ(i)=i=1nai. (1)

Proof.  If  n=1,  we have nothing to prove.  Make the induction hypothesis, that (1) is true for  n=m-1.  Denote




where am has been moved to the end by the induction hypothesis.  But the productMathworldPlanetmathPlanetmath in the parenthesis, which exactly the factors a1,a2,,am-1 in a certain , is also by the induction hypothesis equal to i=1m-1ai.  Thus we obtain


whence (1) is true for  n=m.

Note.  There is mentionned in the Remark of the entry “” a more general notion of commutativity.

Title general commutativity
Canonical name GeneralCommutativity
Date of creation 2014-05-10 21:59:41
Last modified on 2014-05-10 21:59:41
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 10
Author pahio (2872)
Entry type Theorem
Classification msc 20-00
Related topic CommutativeLanguage
Related topic GeneralAssociativity
Related topic AbelianGroup2