general commutativity
Theorem. If the binary operation “” on the set is commutative, then for each in and for each permutation on , one has
(1) |
Proof. If , we have nothing to prove. Make the induction hypothesis, that (1) is true for . Denote
Then
where has been moved to the end by the induction hypothesis. But the product in the parenthesis, which exactly the factors in a certain , is also by the induction hypothesis equal to . Thus we obtain
whence (1) is true for .
Note. There is mentionned in the Remark of the entry “http://planetmath.org/node/2148commutativity” a more general notion of commutativity.
Title | general commutativity |
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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 |