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MODERN ALGEBRA

V. SUMATHI,

DEPARTMENT OF MATHEMATICS,

Dr. R.A.N.M Arts and Science College,

Erode.

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Equivalence Relation

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Partial order

A relation R defined on a set S is said to antisymmeric if a R b and b R a implies a=b. A relation R in S Which is reflexive,antisymmetric and transitive is called a partial order on S.

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A set S with a partial order R defined on it is called a partially ordered or a poset and is denoted by (S,R).

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Functions

Consider the function f:R→R given by f(x)=3 is called a constant function.

A function f:A→B is one -one function if distinct elements are in A have distinct image in B under f.

A function f:A→B is onto function if the range of f is equal to B.

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Binary operations

Ex:

The usual addition, + is a binary operation on N,Z,Q,R and C.

In any non empty set A,* defined by a*b=a is a binary operation.

In(N,*) defined by a*b=aab is a binary operations.

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Permutation Groups

Results

Any permutation can be expressed as a product of disjoint cycle.

Any permutation can be expressed as a product of transpositions.

The product of two even permutations is an even permutations.

The product of two odd permutations is an even permutations.

The product of an even permutations and an odd permutations is an odd permutations.

The inverse of an odd permutations is an odd permutations.

The identity permutations e is an even permutations.

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Cyclic Groups and Order of an element

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Cosets and Lagrange’s Theorem�

Definition:

Let H be a subgroup of a group G. Let a∊G. Then the set aH={ah/h∊H} is called the left coset of H defined by a in G. Similarly Ha={ha/h∊H} is called the right coset of H defined by a.

Result:

Let G be a finite group of order n and H be any subgroup of g. Then the order of h divides the order of G.

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Euler’s Theorem

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Thank you