Groups Algebraic structures
(Monday, May 3, 2010)
A group is a non-empty set G under the binary operation * of the following four axioms are satisfied.
G1. Closure property:
" a, b e G, a*beG
G2. Associative law:
" a, b, c e G, (a*b)*c= a*(b*c)
G3. Existence of right and left identity:
There exists an element e e G,
such that " a e G,
a * e = a (right identity)
e * a = a (left identity)
G4. Existence of right and left inverse
element:
" a e G, there exists an element a' e G,
such that
a * a' = e (right identity)
a' * a = e (left identity)
Note: Existence of identity element in G3 and existence of inverse of an element in G4 occur as left identity in G3 and left inverse The Algebraic Structure of Group Rings (Pure & Applied Mathematics)
Abstract Algebra: Mathematics, Algebraic structure, Group (mathematics), Ring (mathematics), Field (mathematics), Module (mathematics), Vector space, Algebra over a field, Real number, Complex number
Abstract Algebra: Mathematics, Algebraic structure, Group (mathematics), Ring (mathematics), Field (mathematics), Module (mathematics), Vector space, Algebra over a field, Real number, Complex numberAlgebraic Structure of Pseudocompact Groups (Memoirs of the American Mathematical Society)
Algebraic structures: Some aspects of group structure (Mathematical studies;no.2)
Algebraic Structures: Some Aspects of Group Structure
in G4 or right identity in G3 and right inverse in G4, but not in any manner. If in an algebraic structure áG, *ñ, axioms G1 and G2 hold, e be a left identify and each element a e G may posses a right inverse, then G may fail to be a group under the binary operation *.
Abstract Algebra: Mathematics, Algebraic structure, Group (mathematics), Ring (mathematics), Field (mathematics), Module (mathematics), Vector space, Algebra over a field, Real number, Complex numberAbstract Algebra: Mathematics, Algebraic structure, Group (mathematics), Ring (mathematics), Field (mathematics), Module (mathematics), Vector space, Algebra over a field, Real number, Complex numberAlgebraic Structure of Pseudocompact Groups (Memoirs of the American Mathematical Society)Algebraic structures: Some aspects of group structure (Mathematical studies;no.2)Algebraic Structures: Some Aspects of Group Structurein G4 or right identity in G3 and right inverse in G4, but not in any manner. If in an algebraic structure áG, *ñ, axioms G1 and G2 hold, e be a left identify and each element a e G may posses a right inverse, then G may fail to be a group under the binary operation *. Similarly, G having right identity and left inverse for each of its elements may fail to be a group.
A group with binary operation * is denoted by áG, *ñ. In particular the group áG, +ñ is called an additive group, as the binary operation being addition. The group áG, .ñ is called a multiplicative group, as the binary operation being multiplication.
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