Chapter1 - Chapter 1 Group Fundamentals 1.1 1.1.1 Groups...

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Chapter 1 Group Fundamentals 1.1 Groups and Subgroups 1.1.1 Defnition A group is a nonempty set G on which there is defned a binary operation ( a,b ) ab satisFying the Following properties. Closure : IF a and b belong to G , then ab is also in G ; Associativity : a ( bc )=( ab ) c For all a,b,c G ; Identity : There is an element 1 G such that a 1=1 a = a For all a in G ; Inverse : IF a is in G , then there is an element a 1 in G such that aa 1 = a 1 a =1. A group G is abelian iF the binary operation is commutative, i.e., ab = ba For all in G . In this case the binary operation is oFten written additively (( ) a + b ), with the identity written as 0 rather than 1. There are some very Familiar examples oF abelian groups under addition, namely the integers Z , the rationals Q , the real numbers R , the complex numers C , and the integers Z m modulo m . Nonabelian groups will begin to appear in the next section. The associative law generalizes to products oF any fnite number oF elements, For exam- ple, ( ab )( cde )= a ( bcd ) e . A Formal prooF can be given by induction. IF two people A and B Form a 1 ··· a n in di±erent ways, the last multiplication perFormed by A might look like ( a 1 a i )( a i +1 a n ), and the last multiplication by B might be ( a 1 a j )( a j +1 a n ). But iF (without loss oF generality) i<j , then (induction hypothesis) ( a 1 a j a 1 a i )( a i +1 a j ) and ( a i +1 a n a i +1 a j )( a j +1 a n ) . By the n = 3 case, i.e., the associative law as stated in the defnition oF a group, the products computed by A and B are the same. 1
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2 CHAPTER 1. GROUP FUNDAMENTALS The identity is unique (1 0 =1 0 1 = 1), as is the inverse of any given element (if b and b 0 are inverses of a , then b b =( b 0 a ) b = b 0 ( ab )= b 0 1= b 0 ). Exactly the same argument shows that if b is a right inverse, and b 0 a left inverse, of a , then b = b 0 . 1.1.2 Defnitions and Comments A subgroup H of a group G is a nonempty subset of G that forms a group under the binary operation of G . Equivalently, H is a nonempty subset of G such that if a and b belong to H ,sodoes ab 1 . (Note that 1 = aa 1 H ; also, ab = a (( b 1 ) 1 ) H .) If A is any subset of a group G , the subgroup generated by A is the smallest subgroup containing A , often denoted by h A i . Formally, h A i is the intersection of all subgroups containing A . More explicitly, h A i consists of all ±nite products a 1 ··· a n , n , 2 ,... , where for each i , either a i or a 1 i belongs to A . To see this, note that all such products belong to any subgroup containing A , and the collection of all such products forms a subgroup. In checking that the inverse of an element of h A i also belongs to h A i ,weuse the fact that ( a 1 a n ) 1 = a 1 n a 1 1 which is veri±ed directly: ( a 1 a n )( a 1 n a 1 1 )=1. 1.1.3 Defnitions and Comments The groups G 1 and G 2 are said to be isomorphic if there is a bijection f : G 1 G 2 that preserves the group operation, in other words, f ( ab f ( a ) f ( b ). Isomorphic groups are essentially the same; they di²er only notationally. Here is a simple example. A group G is cyclic if G is generated by a single element: G = h a i . A ±nite cyclic group generated
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Chapter1 - Chapter 1 Group Fundamentals 1.1 1.1.1 Groups...

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