Chap2-notes-A

Chap2-notes-A - CHAPTER 2 43 1 Groups 1.1 Basic Concepts of...

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1 Groups 1.1 Basic Concepts of Groups A group is a set G together with a binary operation defned on G such that the Following axioms (conditions) are satisfed: 1. The binary operation is associative. 2. G contains an element e such that, For any element a oF G , a e = e a = a. This element e is called an identity element oF G with re- spect to the operation . 3. ±or any element a in G , there exists an element a 0 in G such that a a 0 = a 0 a = e. The element a 0 is called an inverse oF a , and vice versa, with respect to the operation . A group G is said to be commutative (an abelian ) iF the binary operation defned on it is also commutative. 43 CHAPTER 2
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Theorem 1. The identity element e of a group G is unique . Proof. Suppose both e and e 0 are identity elements of G . Then e e 0 = e and e e 0 = e 0 . This implies that e and e 0 are identical. Therefore, the identity of a group is unique. Theorem 2. The inverse of any element in a group G is unique . Proof. Let a be an element of G . Suppose a has two inverses, say a 0 and a 0 . Then a 0 = e a 0 =( a 0 a ) a 0 = a 0 ( a a 0 )= a 0 e = a 0 . This implies that a 0 and a 0 are identical and hence a has a unique inverse. The set of all rational numbers with real addition + forms a commutative group. The number of elements in a Fnite group is called the order of the group. It is clear that the order of a Fnite group G is simply its cardinality | G | . 44
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Example 1. This example gives a simple group with two elements. We start with the set of two integers, G = { 0 , 1 } . DeFne a binary operation, denoted by ,on G as shown by the Cayley table below: 01 0 1 10 ±rom the table, we readily see that 0 0=0 , 0 1=1 , 1 0=1 , 1 1=0 . It is clear that the set G = { 0 , 1 } is closed under the operation and is commutative. To prove that is associative, we simply determine ( a b ) c and a ( b c ) for eight possible combinations of a , b , and c with a , b , and c in G = { 0 , 1 } , and show that ( a b ) c = a ( b c ) , for each combination of a , b , and c . ±rom the table, we see that 0 is the identity element, the inverse of 0 is itself and the inverse of 1 is also itself. Thus, G = { 0 , 1 } with operation is a commutative group of order 2 . The binary operation deFned on the set G = { 0 , 1 } by the above Cayley table is called modulo-2 addition . 45
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1.2 Finite Groups We present the two most obvious fnite group: one whose operation (on the integers) is addition modulo- m and one whose operation is multiplication modulo- m . Mod- m Addition Let m be a positive integer and consider the set oF integers G = { 0 , 1 ,... ,m 1 } . Defne ¢ as Follows: ±or any two integers i and j in G , i ¢ j = r, (1) where r is the remainder resulting From dividing the sum i + j by m . By Euclid’s division algorithm , r is a non-negative integer be- tween 0 and m 1 and is thereFore an element in G .
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Chap2-notes-A - CHAPTER 2 43 1 Groups 1.1 Basic Concepts of...

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