The Journal of the Indian Mathematical Society

T. Venkatarayudu ¹

Affiliations
1 Andhra University, IN

Abstract

D. E. Littlewood showed that the quickest method of calculating the characters of the symmetric groups is by the use of the recurrence relations between the S-functions. He has given the tables of characters for the symmetric groups up to degree 10.

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R. Vaidyanathaswamy , T. Venkatarayudu

Abstract

Let A be a commutative ring with unit element and A[x] denote the ring of polynomials in x with coefficients from A. Hilbert's theorem is : -

If every ideal in A has a finite basis, then every ideal in A[x] also has a finite basis.

The proof of this as given by Van der Waerden in his Moderne Algebra Bd. II gives us no information beyond the fact that every ideal in A [x] has a finite basis. If we just reverse the argument in his proof we will be able to give actually a canonical basis for every ideal in A[x] which will be found to be a powerful tool in several applications.

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T. Venkatarayudu ¹

Affiliations
1 Andhra University, Waltair, IN

Abstract

The main object of the present paper is to show that the algebra of the Gaussian periods is identical with the algebra of the e^th power residues mod p where p is a prime and e is a divisor of p-1. This algebra is obviously a particular case of the algebra of the e^th power residues mod N where N is any integer. Since the system of residue classes mod N is abstractly identical with a cyclic group of order N, the above algebra may be interpreted to be the algebra of the e^th power residues connected with a cyclic group of order N. It is therefore natural to enquire whether there is a similar algebra connected with any finite Abelian group. We shall however show by means of an example that this is not the case.

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T. Venkatarayudu ¹

Affiliations
1 University of Madras, IN

Abstract

A set of elements is said to form a group under a composition rule R or simply an R-group if

(i) for every two elements a and b of the set aRb is also an element of the set, i.e. the set is closed under R;

(ii) R is associative, i.e. for any three elements a,b,c, of the set (aRb)Rc=aR(bRc);

(iii) there exists an element e, called the identity element, such that for every element a of the set ake=eRa=a;

(iv) to every element a, there exists an element x=a^-1 called the inverse of a, such that aRx=e. The group is called Abelian if R is commutative.

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T. Venkatarayudu ¹

Affiliations
1 University of Madras, IN

Abstract

An arithmetic function f(N) is multiplicative if

f(MN)=f(M)f(N)

whenever the integers M, N are relatively prime. The function F(N) defined by the equation

F(N)=Σf₁(δ)f₂(N/δ)

summed for all divisors δ of N is called the composite of the two arithmetic functions f₁, f₂. If f₁ and f₂ are multiplicative, it is easy to see that their composite F (represented by f₁, f₂) is also multiplicative.

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T. Venkatarayudu ¹

Affiliations
1 University of Madras, IN

Abstract

Let M₁, M₂,..., M_n be a set of n integers. The residue classes with respect to any integer M_r(r=1, 2,......., n) form a ring in which the sum and the product of two residue classes m₁ mod(M_r), m₂ mod (M_r) are (m₁+m₂) mod(M_r) and m₁m₂ mod(M_r) respectively.

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