Group:Let G be a nonvoid set with a binary operation * that assigns to each ordered pair (a, b) of elements of G an element of G denoted by a * b. We say that G is a group under the binary operation * if the following three properties are satisfied: 1) Associativity: The binary operation * is associative i.e. a*(b*c)=(a*b)*c , ∀ a,b,c ∈ G 2) Identity: There is an element e, called the identity, in G, such that a*e=e*a=a, ∀ a ∈ G 3) Inverse: For each element a in G, there is an element b in G, called an inverse of a such that a*b=b*a=e, ∀ a, b ∈ G Note: If a group has the property that a*b=b*a i.e., commutative law holds then the group is called an abelian.Properties of Groups:The following theorems can understand the elementary features of Groups: Theorem1:1. Statement:  In a Group G, there is only one identity element (uniqueness of identity) Proof:  let e and e' are two identities in G and let a ∈ G ∴ ae = a ⟶(i) R.H.S of (i) and (ii) are equal ⇒ae =ae' Thus by the left cancellation law, we obtain e= e' There is only one identity element in G for any a ∈ G. Hence the theorem is proved. 2. Statement:  For each element a in a group G, there is a unique element b in G such that ab= ba=e (uniqueness if inverses) Proof:  let b and c are both inverses of a a∈ G Then ab = e and ac = e Hence inverse of a G is unique. Theorem 2:1. Statement:  In a Group G,(a^{1})^{1}=a,∀ a∈ G Proof: We have a a^{1}=a^{1} a=e Where e is the identity element of G Thus a is inverse of a^{1}∈ G i.e., (a^{1})^{1}=a,∀ a∈ G 2. Statement: In a Group G,(a b^{1})=b^{1} a^{1},∀ a,b∈ G Proof:  By associatively we have (b^{1} a^{1)}ab=b^{1} (a^{1} a)b Similarly (ab) (b^{1} a^{1})=a(b b^{1}) a^{1} Hence the theorem is proved. Theorem3:In a group G, the left and right cancellation laws hold i.e. (i) ab = ac implies b=c (ii) ba=ca implies b=c Proof (i) Let ab=ac (ii) Let ba=ca Hence the theorem is proved. Finite and Infinite Group:A group (G, *) is called a finite group if G is a finite set. A group (G, *) is called a infinite group if G is an infinite set. Example1: The group (I, +) is an infinite group as the set I of integers is an infinite set. Example2: The group G = {1, 2, 3, 4, 5, 6, 7} under multiplication modulo 8 is a finite group as the set G is a finite set. Order of Group:The order of the group G is the number of elements in the group G. It is denoted by G. A group of order 1 has only the identity element, i.e., ({e} *). A group of order 2 has two elements, i.e., one identity element and one some other element. Example1: Let ({e, x}, *) be a group of order 2. The table of operation is shown in fig:
The group of order 3 has three elements i.e., one identity element and two other elements.
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