Nonexistence of Abelian Difference Sets: Lander’s Conjecture for Prime Power Orders
نویسنده
چکیده
In 1963 Ryser conjectured that there are no circulant Hadamard matrices of order > 4 and no cyclic difference sets whose order is not coprime to the group order. These conjectures are special cases of Lander’s conjecture which asserts that there is no abelian group with a cyclic Sylow p-subgroup containing a difference set of order divisible by p. We verify Lander’s conjecture for all difference sets whose order is a power of a prime greater than 3.
منابع مشابه
Some Restrictions on Orders of Abelian Planar Difference Sets
The Prime Power Conjecture (PPC) states that abelian planar difference sets of order n exist only for n a prime power. Lander and others have shown that orders divisible by certain composites can be eliminated. In this paper we show how to extend this list of excluded orders.
متن کاملOn non-existence of some difference sets
Eric Lander conjectured that if G is an abelian group of order v containing a difference set of order n and p is a prime dividing v and n, then the Sylow p-subgroup of G cannot be cyclic. This paper verifies a version of this conjecture for k < 6500. A special case of this version is the non-existence of Menon-Hadamard-McFarland difference sets in 2-groups. We also give an algorithm that easily...
متن کاملThe Prime Power Conjecture is True for
The Prime Power Conjecture (PPC) states that abelian planar difference sets of order n exist only for n a prime power. Evans and Mann [1] verified this for cyclic difference sets for n ≤ 1600. In this paper we use the Mann test to verify the PPC for n ≤ 2, 000, 000.
متن کاملExponent Bounds for a Family of Abelian Difference Sets
Which groups G contain difference sets with the parameters (v, k, A.)= (q 3 + 2q 2 , q2 + q, q), where q is a power of a prime p? Constructions of K. Takeuchi, R.L. McFarland, and J.F. Dillon together yield difference sets with these parameters if G contains an elementary abelian group of order q2 in its center. A result of R.J. Turyn implies that if G is abelian and p is self-conjugate modulo ...
متن کامل2-recognizability of the simple groups $B_n(3)$ and $C_n(3)$ by prime graph
Let $G$ be a finite group and let $GK(G)$ be the prime graph of $G$. We assume that $ngeqslant 5 $ is an odd number. In this paper, we show that the simple groups $B_n(3)$ and $C_n(3)$ are 2-recognizable by their prime graphs. As consequences of the result, the characterizability of the groups $B_n(3)$ and $C_n(3)$ by their spectra and by the set of orders of maximal abelian subgroups are ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2004