preliminaries , binary operations , algebric structures, Schemes and Mind Maps of Abstract Algebra

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Doc. Dr. Miinewver Pinar EROGLU Department of Mathematics Dokuz Eyliil University-lzmir Spring Semester 2024-2025 @ Preliminaries: An overview of key topics covered in the Fundamentals of Mathematics course. @ Binary Operations @ Examples of Binary Operations @ Definitions: Associativity, commutativity, identity, and invertibility. © Algebraic Structures @ Semigroups and Monoids @ Groups @ Basic Examples of Groups @ General Linear Group @® Properties of Groups @ Additional Examples of Groups: the Klein 4-group Oo o = = YA 5. If p is a prime that divides ab, then p divides a or p divides b. 6. Fundamental Theorem of Arithmetic: Every integer greater than 1 is a prime or a product of primes. This product is unique, except for the order in which the factors appear. That is, if N= Pi Po... Pr and N= 4) 9.--s, where the p’s and q’s are primes, then r=s and, after renumbering the g’s, we have p; = q; for all i. 7. The least common multiple of two nonzero integers a and b is the smallest positive integer that is a multiple of both a and b. We will denote this integer by /cm(a, b). 8. Let a,be N. Then a-b = gcd(a, b) -Icm(a, b). 9. A natural number n is called composite if it can be factored as n= ab, where a, b> 1. 10. IT, Principle of Mathematical Induction: Mathematical induction is a proof technique used to establish the truth of an infinite number of statements, typically involving natural numbers. It consists of two steps: the base case, where the statement is proven true for the initial value (usually n= 1); and the inductive step, where it is assumed true for some arbitrary integer n= k, and then proven true for n= k +1, thereby confirming the statement holds for all natural numbers greater than or equal to the base case. Let A,B be sets. The set of all ordered pairs (x,y), where xe A and y €B, is called the Cartesian Product of A and B, in that order, and is denoted by Ax B. In symbols, Ax B= (xy) |xeA, ye Bh. 13. Functions: See the lecture notes on "Functions" covered in the Fundamentals of Mathematics course. 14. Recall the following sets of numbers that we used regularly before: 1. the set of natural numbers {1,2,3,...} denoted N; 2. the set of integers {...,-2,-1,0,1,2,...} denoted Z; 3. the set of rational numbers, denoted Q, which is the set of fractions; 4. the set of real numbers, denoted R, which is the set of all the numbers that are informally thought of as forming the number line; 5. the set of even integers {...,-4,-2,0,2,4,...} denoted E; 6. the set of odd integers {...,-5,-3,-1,1,3,5,...} denoted O. Let us consider the ordinary addition, subtraction and multiplication of numbers on the sets above. And we will check which one defines a function. For instance, we consider the relation f : N x N > N defined by the rule f((a, b)) := a+b. Does this relation define a function? ... (we will discuss the outcomes and more in class) ... qo > «

«Erp a Br = Ae a) The sum of any two natural numbers is a natural number, and hence we can think of addition on N as a function +:NxN-N, which means that addition is a binary operation on N. b) Subtraction is not a binary operation on N, because the difference of two natural numbers is not always a natural number. However, subtraction is a binary operation on Z. c) The most familiar binary operations are ordinary addition, subtraction and multiplication of integers. We can also replace Z with any of the sets Q and R. d) Division of integers is not a binary operation on the integers because an integer divided by an integer need not be an integer. e) Is addition on E (resp. O) a binary operation? oO ra = f) Just as multiplication of numbers is often taught by using multiplication tables, we can define binary operations on finite sets by using operation tables. For example, let A = {a,b,c}. We define two binary operations * and o on A by the operation tables: *|a b ¢ ola be ala bc a|b aa b|b ba blac b ¢|c 6b 2 cla bec Similarly, you can define another binary operation on A. The question then becomes: how many distinct operations can be defined on A? 1. A binary operation * on a set S is commutative if ax b= px og for all elements a and bin S. 2. A binary operation * on a set S is associative if (ax b)*c=ax(b*c) for all elements a,b and cin S. 3. Let S be a set with a binary operation +. An element e in S is called an identity element (or just an identity) if e*a=a and axe=a for every element ain S. Ex. 1. Determine whether the binary operations in examples (a) through (i) are commutative and associative. Focus only on the binary operations for your analysis. a) The ordinary addition on N is a commutative and associative. b) . ey: Ex. 2. Determine whether the binary operations in examples (a) through (i) above have an identity element or not. Focus only on the binary operations for your analysis. Let S be a set with a binary operation * and an identity element e. e Let a be an element in S. If there exists an element bin S such that ax b=e=b-* a) then the element a is called invertible, and b is called an inverse of a. In this case, we write itas b=a"?. We notice that in order to check whether an element its invertible in S, the set S must have an identity element under the binary operation on S. Furthermore, by the definition, it is possible for some elements to be invertible while others are not. Ex. 3. Let's determine whether any elements in examples (a) through (i) are invertible. Notice that we only check binary operations with an identity element. a)... by se c) Addition on Z is a binary operation with 0 serving as the identity element. Therefore, we can look for invertible elements in Z under addition. Thus, is 1 invertible? If it is, what is the inverse of 1? Is every element 2€ Z invertible under addition? e A set S with a binary operation » is called a semigroup if * is associative. e A semigroup (S, *) is called a monoid if it has an identity element. e Thus, every example in (a) through (i) that is associative is a semigroup. Moreover, those that have an identity element are monoids. e Furtherfore, x+y Z defines a binary operation on Q which is not associative. Thus, (Q, *) is an algebraic group but not a semigroup. Xe VIS oO A monoid (G, *) is called a group if every element of G has an inverse. Therefore, we say an algebraic structure (G,*) is a group if the following three properties are satisfied. 1. Associativity. The operation * is associative, 1.é., (ax b)*c=a*(bxc) for all elements a,b and c in G. 2. Identity. There is an identity element e in G, i.e., ax b=e=b* a, for every element a in G. 3. Inverses. For each element a in G, there is an element bin G such that a+ b=e= 5» a, namely, a is invertible. Dee... q o = = = )ac