Saturday, December 31, 2016

Reduced Abelian Groups

  I'm having a little trouble classifying my operators for the magic squares. For one thing the left and right products are "duals" of each other. Groups usually have only one product. In Combinatorial Group Theory they talk about words, factors and primitive elements but they might call FG a primitive element. Abelian groups can be reduced. Transpose is a complication but can be treated as second magic square. The "class" defined by F, G and transpose may be extensible under some unique circumstances. For instance one can go from one of the transformed simple magic squares to Dürer's by exchanging the two middle rows because the equal sums are 17. We first showed that the second diagonal flip could be represented by a transpose and multiplication by F. Then we showed that we could do something similar with the rotations. There may be room for improvement in group theory.

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