Probability that the commutator of two group elements is equal to a given element

AbstractIn this paper we study the probability that the commutator of two randomly chosen elements in a finite group is equal to a given element of that group. Explicit computations are obtained for groups G which |G′| is prime and G′≤Z(G) as well as for groups G which |G′| is prime and G′∩Z(G)=1. This paper extends results of Rusin [see D.J. Rusin, What is the probability that two elements of a finite group commute? Pacific J. Math. 82 (1) (1979) 237–247]

Zero-divisor graph with respect to an ideal

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A note on the countable union of prime submodules

Let M be a finitely-generated module over a Noetherian ring R. Suppose is an ideal of R and let N=M and =Ann(M/N). If ⫅J(R), M is complete with respect to the -adic topology, {Pi}i≥1 is a countable family of prime submodules of M, and x∈M, then x+N⫅∪i≥1Pi implies that x+N⫅Pj for some i≥1. This extends a theorem of Sharp and Vámos concerning prime ideals to prime submodules

La Poliomyelite en Iran Premieres Recherches de Laboratoire

This research work on the status of poliomyelitis in Iran, performed in 1954, in 1958-59, consisted in providing data on the past history of the infection and on its present diffusion in this country. The serological and COprologi~al surveys produced evidence of conditions similar to thos.e of other countries .WIth the same public health situation: large and permanent (WIthout seasonal v~riatlOnSJ diffusion of the e~teroviruses, reaching percentages of 37 to 74% (accol~'dmg to the groups) of ca:rri!ers-excreters in a sampling of healthy chi~dren u~der five y~a!'s of age, providing high levels of early and complete Immunlzation against the pohomyelitic disease

3-Designs with Block Size 6 from PSL(2, q) and Their Large Sets

We investigate the existence of 3-designs and uniform large sets of 3-designs with block size 6 admitting PSL(2, q) as an automorphism group. We show the existence of simple 3-(28, 6, 10m) designs for 1 ≤ m ≤ 230. Most of these designs were previously unknown