70 1. ALGEBRAIC THEMES Examples we have considered so far are as follows: ± The set of symmetries of a geometric ﬁgure with composition of symmetries as the product. ± The set of permutations of a (ﬁnite) set, with composition of per-mutations as the product. ± The set of integers with addition as the operation. ± Z n with addition as the operation. Indeed, Proposition 1.7.7 , parts (a), (b), and (c), says that addition in Z n is associative and has an identity Œ0Ł , and that all elements of Z n have an additive inverse. ± KŒxŁ with addition as the operation. In fact, Proposition 1.8.2 , parts (a), (b), and (c), says that addition in KŒxŁ is associative, that0 is an identity element for addition, and that all elements of KŒxŁ have an additive inverse. It is convenient and fruitful to make a concept out of the common char-acteristics of these several examples. So we make the following deﬁnition: Deﬁnition 1.10.1. A group is a (nonempty) set G with a product, denoted here simply by juxtaposition, satisfying the following properties:
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.