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Groups| Modern Abstract Algebra

Definition of group: A group is a non empty set G together with a binary operation multiplication (*) such that the following four properties are satisfied

(i) Closure Property: a*b∊ G, ∀ a, b ∊G

(ii) Associative property: (a * b) * c = a * ( b* c), ∀ a, b, c ∊G

(iii) Identity property: There exist an element e ∊G called an identity or unit element such that, a * e = e * a = a, ∀ a ∊ G.

(iv) Inverse property: For each a ∊ G there exist an element b ∊ G such that a * b = e = b * a.

Abelian groups: A group (G; *) is abelian or commutative if the binary operation * is commutative.

Finite group and infinite group: A group G is said to be finite group or an infinite group according as it contains a finite or an infinite number of elements.

Order group: The order of a group G, denoted by o(G), |G|, is the number of elements in G.

Additive and multiplicative group: A group (G, *) is called an additive group or a multiplicative group according as its binary operation * is addition or multiplication.

Thus (Z, +) is an additive group while (R+,×) is a multiplicative group.

Uniqueness of identity and inverse in a group: If (G, *) is a group, then (i) G has a unique identity (ii) Every a ∊ G has a unique inverse in G.

Proof: (i) If possible, let G have two identities e and f.

Then for every x ∊ G.

x * e = e * x = x ——————–(i)

and x * f = f * x = x —————-(ii)

Putting x = f in (i) we have, f * e = e * f = f

Putting x = e in (ii) we have, e * f = f * e = f

Thus e = f * e = e * f = f

This proves the uniqueness of identity.

(ii) Let a ∊ G have two inverse say x and y ∊ G

Then a * x = x * a = e and a * y = y * a = e

Now x = x * e = x * ( a * y), using a * y = e

= (x * a) * y, by associatively

= e * y = y using x * e =e

Hence has a unique inverse.

Groupoid: A Groupoid is a non empty set S with a binary operation * such that S is closed under multiplication.

i.e., a * b ∊ S, ∀ a, b ∊S

Semi group: A groupoid (S, *) is called a semi group or quasi group, if S is associative for *.

Loop: A groupoid (S, *) is called a loop if S has the identity element e for *,

E * x = x * e = x, ∀ x ∊ S.

Monoid: A semi group (S, *) is called a monoid if S contains the identity e for *;

e * x = x * e = x, ∀ x ∊ S.

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