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Unformatted text preview: 1.10. GROUPS 71 and 0s off the diagonal, is the identity element for matrix mul tiplication, and the inverse of a matrix is the inverse for matrix multiplication. The Virtues of Abstraction Abstraction , the process of elevating recurring phenomena to a con cept, has several purposes and advantages. First, it is a tool for organizing our knowledge of phenomena. To understand phenomena, it already helps to classify them according to their common features. Second, abstraction yields efficiency by eliminating the need for redundant arguments; certain results are valid for all groups, or for all finite groups, or for all groups satisfying some additional hypothesis. Instead of of proving these results again and again for symmetry groups, for permutation groups, for groups of invertible matrices, and so on, we can prove them once and for all for all groups (or for all finite groups, or for all groups satisfying property xyz). Indeed, finding more or less the same arguments employed to esxyz)....
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 Fall '08
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 Algebra, Multiplication

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