2 1. Vecto r Spaces and is often omitted altogether. We also simplify the notation by refer- ring to the group, when the binary operation is assumed known, as G, rather than (G, *). EXAMPLES: a. (Z , +), the integers with standard addition. b. (E \ {0} , •) , th e nonzero real numbers, wit h standar d multiplica - tion. c. S n — (Sn, •) , the symmetric group on[l,...,n] «a positiv e integer. The elements of Sn ar e all the permutations a o f the set [1,... , n], i.e., the set of the bijections (1-1 maps) of [1,.. . ,n] onto itself. The group operation is composition: for a , T G Sn w e define TO by: (T • a)(j) = *(./)) f o r a1 1 J in t 1 ^]- The first two examples are commutative the third is not if n 2. 1.1.2 Fields . DEFINITION: A field (F,+,») i s a set F endowe d with two binary operations, addition, (a,b) \-^ a + b9 an d multiplication, (a,b) \- a-b, (usually written simply as ab), such that: F-1 (F , +) is a commutative group, whose identity element is denoted byO. F-2 (F \ {0},- ) i s a commutative group , whos e identit y elemen t i s denoted by 1. It is the multiplicative group of F. F-3 Additio n and multiplication are related by the distributive law. a(b + c) — ab + ac. EXAMPLES: a. Q , the field of rational numbers. b. R , the field of real numbers.
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