Addition Modulo
Now we are going to discuss a new type of addition which is known as “addition modulo m” and written in the form
where a and b belongs to an integer and m is any fixed positive integer.By definition we have
Addition & Multiplication of Modulo - Abstract Algebra
where a and b belongs to an integer and m is any fixed positive integer.By definition we have
(0)
Properties of Group
- The identity element of a group is always unique.
- The inverse of each element of a group is unique, i.e., in a group G with operation * for every
, there is only element 
such that
, e being the identity.
Table for Group- Abstract Algebra
The composition tables are useful in examining the following axioms in the manner explained below:
- Closure Axiom : If all the elements of the table belong to the set G (say) then G is closed under the Composition a (say). If any of the elements of the table does not belong to the set, the set is not closed.
Composition Table - Abstract Algebra
A Binary Operation in a finite set can completely be described with the help of a table. This table is well known as composition table. The composition table helps us to verify most of the properties satisfied by the binary operations. This table can be formed as follows:
Order of a Group- Abstract Algebra
Finite and infinite Groups:
If a group contains a finite number of distinct elements, it is called finite group otherwise an infinite group.
In other words, a group
is said to be finite or infinite according as the underlying set G is finite or infinite.
In other words, a group
is said to be finite or infinite according as the underlying set G is finite or infinite. Commutative Group or Abelian Group
If the commutative law holds in a group, then such a group is called an Abelian group or Commutative group. Thus the group
is said to be an Abelian group or commutative group if
, 
.
A group which is not Abelian is called a non-Abelian group. The group 
is called the group under addition while the group 
is known as group under multiplication.
is called the group under addition while the group is known as group under multiplication.
Algebraic Structure- Abstract Algebra
A non-empty set 
together with at least one binary operation defined on it is called an algebraic structure. Thus if G is a non-empty set and “
” is a binary operation on
, then 
is an algebraic structure.
are all algebraic structures. Since addition and multiplication are both binary operations on the set R of real numbers,
is an algebraic structure equipped with two operations.
together with at least one binary operation defined on it is called an algebraic structure. Thus if G is a non-empty set and “” is a binary operation on, then is an algebraic structure.are all algebraic structures. Since addition and multiplication are both binary operations on the set R of real numbers,
is an algebraic structure equipped with two operations.Identity and Inverse
Identity: A composition 
in a set
is said to possesses of an identity if there exists an element 
such that
Moreover, the element e, if it exists is called an identity element and the algebraic structure
is said to have an identity element with respect to
.
in a set is said to possesses of an identity if there exists an element such thatMoreover, the element e, if it exists is called an identity element and the algebraic structure
is said to have an identity element with respect to. Binary Operations
The concept of binary operation on a set is a generalization of the standard operations like addition and multiplication on the set of numbers. For instance we know that the operation of addition (+) gives for ally two natural numbers m,n another natural number m+n, similarly the multiplication operation gives for the pair m,n the number m,n in N again. These types of operations arc found to exist in many other sets. Thus we give the following definition.
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