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Algebra: Abstract and Modern by U. M. Swamy, A. V. S. N. Murthy

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3

Groups

3.1 BINARY SYSTEMS

It is well known that the product of two integers is again an integer. That is, if a and b are integers, then the product a X b is again an integer. Here, the symbol ‘.’ denotes the ‘operation of taking product’ of a and b, in this order. Similarly, if A and B are two 2 × 2 matrices over the real number system, then the product A . B is again a 2 × 2 matrix over ℝ. Here, the symbol ‘ .’ denotes the ‘operation of taking product’ of the matrices A and B, in this order. Further, if f and g are two mappings of a set X into itself, then the composition g o f is also a mapping of X into itself. Here, the symbol ‘o’ denotes ...

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