Surinder K. Sehgal

New constructions of Menon difference sets

Abstract Menon difference sets have parameters (4N2, 2N2 − N, N2 − N). These have been constructed for N = 2a3b, 0 ⩽ a,b, but the only known constructions in abelian groups require that the Sylow 3-subgroup be elementary abelian (there are some nonabelian examples). This paper provides a construction of difference sets in higher exponent groups, and this provides new examples of perfect binary arrays.

Weak Cayley tables

In [1] Brauer puts forward a series of questions on group representation theory in order to point out areas which were not well understood. One of these, which we denote by (B1), is the following: what information in addition to the character table determines a (finite) group? In previous papers [5, 7–13], the original work of Frobenius on group characters has been re-examined and has shed light on some of Brauers questions, in particular an answer to (B1) has been given as follows. Frobenius defined for each character χ of a group G functions χ(k):G(k) → C for k = 1, …, degχ with χ(1) = χ. These functions are called the k-characters (see [10] or [11] for their definition). The 1-, 2- and 3-characters of the irreducible representations determine a group [7, 8] but the 1- and 2-characters do not [12]. Summaries of this work are given in [11] and [13].

Some new difference sets

Abstract Existence status of (96, 20, 4) difference sets in Z 4 × Z 4 × Z 2 × Z 3 has been open so far. In this paper we construct these difference sets, thereby filling a missing entry in Landers table with the answer “yes,”

An exponent bound on skew Hadamard abelian difference sets

A difference setD in a groupG is called a skew Hadamard difference set (or an antisymmetric difference set) if and only ifG is the disjoint union ofD, D(−1), and {1}, whereD(−1)={d−1|d∈D}. In this note, we obtain an exponent bound for non-elementary abelian groupG which admits a skew Hadamard difference set. This improves the bound obtained previously by Johnsen, Camion and Mann.

On Abelian Difference Sets

We review some existence and nonexistence results — new and old — on abelian difference sets. Recent surveys on difference sets can be found in Arasu (1990), Jungnickel (1992a, b), Pott (1995), Jungnickel and Schmidt (1997), and Davis and Jedwab (1996). Standard references for difference sets are Baumert (1971), Beth et al. (1998), and Lander (1983). This article presents a flavour of the subject, by discussing some selected topics.

The 2-character table does not determine a group

Frobenius had defined the group determinant of a group G which is a polynomial in n = IG( variables. Formanek and Sibley have shown that the group determinant determines the group. Hoehnke and Johnson show that the 3-characters (a part of the group determinant) determine the group. In this paper it is shown that the 2-characters do not determine the group. If we start with a group G of a certain type then a group H with the same 2-character table must form a Brauer pair with G. A complete description of such an H is available in Comm. Algebra 9 (1981), 627-640.

Using the Simplex Code to Construct Relative Difference Sets in2-groups

Relative Difference Sets with the parameters (2a, 2b, 2a, 2a-b) have been constructed many ways (see davis, EB, jung, maschmidt, and pottsurvey for examples). This paper modifies an example found in arasusehgal to construct a family of relative difference sets in 2-groups that gives examples for b = 2 and b = 3 that have a lower rank than previous examples. The Simplex code is used in the construction.

Difference sets in Abelian groups of p -rank two

AbstractUnder a technical assumption that pertains to the so-called “self-conjugacy”, we prove: if an abelian groupG ofp-rank two,p a prime, admits a (nontrivial) (v, k, λ) difference setD, then for each

A Family of Semi-Regular Divisible Difference Sets

x D,x.C_p \subseteq D