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 differenc...

Let $K$ be a subgroup of finite group $G$ . The probability that an element commutes with is denoted by $Pr(K,G)$ Assume $Pr(K,G)\geq \epsilon$ for some fixed $\epsilon >0$ We show there normal $T\leq G$ and $B\leq K$ such the indices $[G:T]$ $[K:B]$ order commutator $[T,B]$ are $\epsilon$ -bounded. This extends well-known theorem, due to P. M. Neumann, covers case where $K=G$ deduce number ...

‎Suppose G is a group of order p^2q^2 where p>q are prime numbers and suppose P and Q are Sylow p-subgroups and Sylow q-subgroups of G, ‎respectively‎. ‎In this paper‎, ‎we show that up to isomorphism‎, ‎there are four groups of order p^2q^2 when Q and P are cyclic‎, ‎three groups when Q is a cyclic and P is an elementary ablian group‎, ‎p^2+3p/2+7 groups when Q is an elementary ablian group an...

In this paper we study finite groups which possess a strongly pembedded subgroup for some prime p. Suppose that p is a prime. A subgroup H of the finite group G is said to be strongly p-embedded in G if the following two conditions hold. (i) H < G and p divides |H|; and (ii) if g ∈ G \H , then p does not divide |H ∩H|. One of the most important properties of strongly p-embedded subgroups is tha...

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 ...

If a group G has a maximal subgroup H, tl-.en G acts by right multiplication on the set fi of right cosets of H in G as a primitive, but not necessarily faithful, permutation group. As part of the search for new finite simple groups it is of interest to determine all possible groups G, and in particular all simple groups, which contain a given group H as a maximal subgroup. This problem is very...

Let p be a prime. A group is called p-closed if it has a normal Sylow p-subgroup and it is called p-exponent closed if the elements of order dividing p form a subgroup. A group is minimal non-p-closed if it is not p-closed but its proper subgroups and homomorphic images are. Similarly, a group is called minimal non-p-exponent closed if it is not p-exponent closed but all its proper subgroups an...

Let E/F be a Galois extension of number fields with Galois group G. The purpose of this paper is to place limitations on the structure of a Sylow 2-subgroup of G in the case when the extension E/F is a minimal counterexample to Artin’s Conjecture on the holomorphy of L-series. More specifically, assume for some s0 ∈ C − {1} and some irreducible character χ of G that the Artin L-series L(s, χ,E/...

Let Gp be a Sylow p-subgroup of the finite groupG and let Char G n (Gp) represent the set of degree n complex characters of Gp that are the restrictions of class functions on G. We construct a natural map ψG : [BG,BU(n)] → ∏ p||G| Char G n (Gp) and prove that ψG is a surjection for all finite groups G that do not contain a subgroup isomorphic to (Z/p) for any prime p. We show, furthermore, that...