کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
4586699 1334110 2009 11 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
On counterexamples to the Hughes conjecture
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات اعداد جبر و تئوری
پیش نمایش صفحه اول مقاله
On counterexamples to the Hughes conjecture
چکیده انگلیسی

In 1957 D.R. Hughes published the following problem in group theory. Let G be a group and p a prime. Define Hp(G) to be the subgroup of G generated by all the elements of G which do not have order p. Is the following conjecture true: either Hp(G)=1, Hp(G)=G, or [G:Hp(G)]=p? After various classes of groups were shown to satisfy the conjecture, G.E. Wall and E.I. Khukhro described counterexamples for p=5,7 and 11. Finite groups which do not satisfy the conjecture, anti-Hughes groups, have interesting properties. We give explicit constructions of a number of anti-Hughes groups via power-commutator presentations, including relatively small examples with orders 546 and 766. It is expected that the conjecture is false for all primes larger than 3. We show that it is false for p=13,17 and 19.

ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Journal of Algebra - Volume 322, Issue 3, 1 August 2009, Pages 791-801