نابخش ‌پذیری گروه‌های آبلی

در سرتاسر متن گروه‌ها آبلی هستند.‌ گروه g را –n بخش‌پذیر گوئیم هرگاه.‌ گروه g را مطلقاً نابخش‌پذیر گوئیم هرگاه برای هر، فاقد زیرگروه ناصفر –n بخش‌پذیر باشد.‌ در بررسی کلاس c متشکل از تمام گروه‌های مطلقاً نابخش‌پذیر مانند g، به زیرگروه‌های جمع تمام زیرگروه‌های p بخش‌پذیر و (برای هر عدد اوّل p) بر می‌خوریم.‌ خواص این دو زیرگروه به تفصیل مورد بررسی قرار گرفته است و برای کلاس تمام گروه‌های  بخش‌پذیر  و کلاس متشکل از تمام گروه‌ها با، ثابت می‌کنیم زوج یک نظریه تاب است.‌ کلاس c تحت هر جمع مستقیم و هر حاصل‌ضرب بسته است و اگر آن‌گاه نشان می‌دهیم.‌ همچنین ثابت می‌شود که اگر و تنها اگر برای هر p، اگر و تنها اگر.‌ سرانجام مشخص‌سازی دیگری برای زیرگروه‌هایی از q (اعداد گویا) که به c تعلق دارند، بیان شده است.‌ مثال‌های متنوع نیز جهت توصیف نتایج آورده شده است.‌ 

non-divisibility of abelian groups

introduction  in throughout all groups are abelian. suppose that g is a group and n is a positive integer. for a ∈ g, if we consider the solution of the equation nx = a in g, two subsets of g are proposed. one of them is {a ∈ g | ∃x ∈ g,nx = a} and the other is {x ∈ g | nx = a} for given a ∈ g. the first is ng, which is clearly a subgroup of g, but the second does not have to be a subgroup. however, if we replace the equation nx = a with nx ∈< a > then we come to the equation nx = 0 in the group, whose solutions determine a subgroup of   (hence of g). in this regard, we state something about divisibility from [2]. let a is an element in a group g. the element a is called divisible whenever for every n ≥ 1 there exists x ∈ g such that nx = a. also a is called torsion whenever there exists positive integer m such that a is a solution of the equation mx = 0. the group g is then called divisible (resp. torsion) if every element in g is divisible (resp. torsion). furthermore, g is called reduced (resp. torsionfree) if it has no nonzero divisible (resp. torsion) subgroup. therefore, g is divisible if and only if ng = g for every n ≥ 1. as canonical examples, we can mention the additive group q and. here, is the subgroup of q/z generated by {1/pi + }. also, and all proper subgroup of q are reduced; [1] and [2] are excellent references on the subject.suppose that n ≥ 1. it is easy to verify that ng = g if and only if pg = g for every prime number p | n. this follows that g is divisible if and only if pg = g for every prime number p. thus g is non − divisible if there exists a prime number q such that qg ≠ g. based on the above, we may define the divisibility (nondivisibility) with respect to a number.definition 1.1. let n ≥ 1 a group g is called:(a)            ndivisible if ng = g.(b)           fully nondivisible if pg ≠ g for every prime number p.(c)            absolutely nondivisible if ph ≠ h for every prime number p and nonzero subgroup h of g.thus, we deal with three class of groups as blow:{absolutely nondivisible groups} ⊆ {fully nondivisible} ∩ {reduced groups}. examples are presented to show that these three classes are mutually distinct.main resultsdefinition 2.1. for every prime number p, let radp(g) = ∩n≥1png and tp(g), the sum of all pdivisible subgroups of g. dn be the class of all ndivisible groups and fp be the class of all groups g with tp(g) = {0}. let d={g|g=h⩽gh such that h ∈ ∪n≥1dn} and cp be the class of all groups g with radp(g) = {0}.theorem 2.2. let p be a prime number.for every group homomorphism f: g1 → g2 we have f (radp(g1)) ⊆ radp(g2). furthermore radp(g) is a fully invariant subgroup of g.for every h⩽g we have radp(h) ⊆ radp(g). also if g = h⊕k then radp(h) = h ∩ radp(g).radp(⊕i∈igi)=⊕i∈iradp(gi).radp(i∈igi)=i∈iradp(gi).pg = g if and only if radp(g) = g if and only if  homz(g,zp)={0}.for every h⩽g we haveradp(g)+hh⊆radp(gh). also, if h⊆radp(g) thenradp(g)h=radp(gh). furthermoregradp(g)={0} .radp(g)=rej(g,cp).radp(g)=rej(g,{zpi}i≥1).theorem 2.3. let p be a prime number.the class of pdivisible groups is closed under direct sum and homomorphic image.for every group g, tp(g) is pdivisible and we have tp(g) ⊆ radp(g). furthermore tp(radp(g)) = radp(tp(g)) = tp(g).if g is a ptorsionfree group, then radp(g) is a pdivisible subgroup and radp(g) = tp(g).let g be a ptorsionfree group and h ≤ g. h ⊆ radp(g) if and only if radp(g) = radp(h). furthermore radp(radp(g)) = radp(g).if  tp(g) = {0}, then p divide the order of every torsion element in g.let p and q be two different prime numbers. if tp(g) = tq(g) = {0}, then radp(g) = radq(g) = {0}.tp(gtp(g))={0}.theorem 2.4. for every prime number p, (dp,fp) is a torsion theory.theorem 2.5. every absolutely nondivisible group g is torsion free and so g is isomorphic to a subgroup ofqλ.theorem 2.6. the following statements are equivalent for every group g.g is absolutely nondivisible,for every prime number p, radq(g) = {0},homz(d,g)={0}.theorem 2.7. the class of absolutely nondivisible is closed under direct product and subgroup.theorem 2.8. if h and g/h are absolutely nondivisible groups then g is absolutely nondivisible group.theorem 2.9. for every group g the following statements hold..(b) g is an absolutely nondivisible group if and only if for every prime number p there exists a natural number n such that  is absolutely nondivisible.for h⩽q and prime number p, letbp(h)={t∈n|∃mn∈h,(m,n)=1,pt|n}, and bp(h) = |bp(h)|.theorem 2.10. let {0} ≠ g ≤ q g is absolutely nondivisible if and only if for every prime number p, bp(g)<∞.