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GROUP THEORY Groupoid ,Semi Group, Monoid, group, Abelian Group, order of a group, properties of a Group , Examples, Order of an element of a group, Modulo System, Permutation group.
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Groupoid Definition; G be a non empty set and ∗ be a binary operation , then the structure (G,*) is called a groupoid, if 𝑎∗𝑏∈𝐺, ∀𝑎,𝑏∈𝐺 𝑖.𝑒 g is closed for the binary operation. Examples : (i)The structures 𝑁,+ , 𝑁,. , 𝑍.+ , 𝑍,. ,( 𝑄,+ , 𝑄,. (ii) The set N is not a groupoid with respect to operation ‘−’ .
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Monoid Definition; Let G be a non empty set and binary operation 𝐺,∗ defined on it, satisfyiny the following axioms; Associative law , Existance of Identity. Example; 𝑍.+ , 𝑍,. , 𝑄,+ , 𝑄,. , 𝑅,+ ,(𝑅,.)
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Group Definition: Let G be a non empty set and ∗ be a binary operation defined on it ,then the structure 𝐺,∗ is said to be a group if the following axioms are satisfied. Clousure, Associativity, Existance of identity, existance of inverse.
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Abelian Group Definition; A group 𝐺,∗ is said to be abelian group or commutative group ,if 𝑎∗𝑏=𝑏∗𝑎 ∀ 𝑎,𝑏∈𝐺. Example ;Set of integers, set (1,−1,𝑖,−𝑖). A group is not simply a set , but it is an algebric structure.
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Order of a group The number of elements in a finite group is called order of a group. It is denoted by 𝑂 𝐺 . An infinite group is called a group of infinite order.
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Order of an Element of a Group
Let G be a group under multiplication. Let e be the identity element in G Suppose, a is any element of g, then the least positive integer n, if exist, such that an = e, is said to be order of an element a € G, and can be written as o ( a) = n In case, such a positive integer n does not exist, we say that the element a is of infinite or zero order ii. Consider the additive group Z= {……,-3,-2,-1, 0, 1, 2, 3,…..} 1.0 = 0 = order of zero is one (finite) but na≠0 is infinite.
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Properties of Groups Let (G,*) be a group, then
The identity element ‘e’ is unique. There exists unique inverse in G.i.e.,a-1 G, A a*c=b*c⇒ a=b ( Right cancellation Law) c* a=c*b⇒ a=b ( Left cancellation Law) Where, a,b,c ∊G The left identity is also right identity i.e., e.a=a and a.e=a The left inverse of an element is also its right inverse i.e., a-1 a=e and aa-1 =e The equation a*x=b and y*a=b, where a, b∊ G have unique solutions in G, which are x=a-1 * b G and y=b * a -1 ∊G, respectively.
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Permutation Groups A one-one mapping f of a finite non-empty set S onto itself is called a permutation. If the set Ѕ consists of n distinct elements, then a one-one mapping of Ѕ onto itself is called a permutation of degree n. Let Ѕ = {a1, a2,…,an}. Then, we denote a permutation f on the set S in a two-rowed notation. So that in the first row all the elements of S are written in certain order and f(a1)=b1,f(a2)=b2……,f(a1)=b1,….,f(an)=bn, Where sare s.
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Types of Permutations Equality of Permutations
Two permutations f and g of a set S are said to be equal, if f(a) = g(a),Ɐ a ∊ S. e.g., if And Of degree 3, then we have f=g since, f(1)= g(1)=3,f(2)=g(2)=1 f(3)=g(3)=2
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Identity Permutation A permutation on the set S is called the identity permutation, it maps each element of S onto itself, It is usually denoted by the symbol l. Thus, l(a)=a,Ɐ a ∊ S e.g, is the identity permutation of degree n.
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Inverse Permutations Since, a permutation is one-one onto mapping and hence, it is invertible, i.e., every permuta- tion f on a set P= {a1, a2,…..an} has a unique inverse permutation denoted by f -1 e.g., If and
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Permutation Group (Symmetric Group)
Let A (S) denote the set of all permutation on a non-empty set S. Then, A(S) form a group under the composition of maps. Moreover, if S contains n elements, then the permutation group A(S) contains n! elements. The group of n! permutations of a set n elements is called symmetric group ( Sn) of degree n. Example 6.Write down all the elements of the permutation group ( or symmetric group) S3 on three elements 1,2 and 3. Solution. Let S= {1, 2, 3} Then, there are 3! = 6 elements in S3. which can be written as
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