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If S has just one element, there is only one possible binary operation on S; the table must be filled in with that single element. The vectors in a subset S of a vector space V over a field F are linearly independent over F if and only if the zero vector cannot be expressed as a linear combination, having some coefficients nonzero, of vectors in S. In particular, one at least need to learn Group Action. This is possibly one of the most elegant books on mathematics I have ever read.

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This should be studied first before moving to the more mature text by Artin.

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