Группы, удовлетворяющие условиям минимальности для ненормальных, абелевых ненормальных, неабелевых ненормальных подгрупп, рассматриваются в настоящей статье

Черников, Н.С.

Repository of Gomel State University

УДК 519.41/47 
 
Groups with some minimal conditions for non-normal subgroups 
 
N.S. CHERNIKOV 
 
The groups, satisfying the minimal conditions for non-normal, abelian non-normal, non-abelian non-
normal subgroups, are considered in the present article. 
Keywords: minimal conditions, non-normal, abelian non-normal, non-abelian non-normal subgroups, 
Chernikov, Artinian, Dedekind, weakly graded groups. 
 
Группы, удовлетворяющие условиям минимальности для ненормальных, абелевых ненормальных, 
неабелевых ненормальных подгрупп, рассматриваются в настоящей статье. 
Ключевые слова: условия минимальности, ненормальные, абелевы ненормальные, неабелевы не-
нормальные подгруппы, черниковские, артиновы, дедекиндовы, слабо ступенчатые группы. 
 
 Remind that a (non-abelian) group with all subgroups normal is called Dedekind (respective-
ly Hamiltonian). The remarkable R. Baer’s Theorem [1] states that the group G is Dedekind iff G is 
abelian or G=Q×E×R with quaternion, elementary abelian 2- and abelian 2′- subgroups Q, E, R re-
spectively. The remarkable S.N. Chernikov’s Theorem [2] states that the group G, having a series 
with finite factors, satisfies the minimal condition for non-normal subgroups (briefly, satisfies 
nmin− ) iff it is Chernikov or Dedekind. Further, remind that the group G is called weakly graded, 
if for g, h ∈G, the subgroup <g, hg > possesses a subgroup of finite index≠1 whenever g is of infinite 
order, or g is a p-element≠1 with some odd prime p and also [ pg ,h]=1 and <g,h> is periodic 
(N.S. Chernikov, see, for instance, [3]). 
 The class of weakly graded groups is very large and includes the classes of binary and locally 
graded, locally finite and locally solvable groups, groups, having a series with finite factors; binary 
finite, linear, 2-groups, periodic Shunkov groups, … . Further, let Ł be the minimal local class of 
groups closed with respect to forming subgroups, series (and, at the same time, to forming subcarte-
sian products) and containing the class of weakly graded groups. The class Ł is extremely large. 
 The following new theorem holds. 
 Theorem 1 (N.S. Chernikov). Let G be a group with nmin− . Then G′ is Artinian. Further, 
G is Chernikov or Dedekind iff G′∈Ł. 
 The Ol’shanskiy’s Examples of infinite simple Artinian groups (see, for instance, [4]) are, of 
course, the examples of non-Chernikov non-Dedekind groups with nmin− . The group is called 
primitive graded, it for g,h∈G, the subgroup <g, hg > possesses a subgroup of finite index ≠1 when-
ever g is a p-element ≠1 with some odd p and also [ pg ,h]=1 and <g,h> is periodic 
(N.S. Chernikov, 2013). The periodic group is primitive graded iff it is weakly graded. 
 Corollary 1 (N.S. Chernikov). The group G with primitive graded G′ satisfies nmin− iff it 
is Chernikov or Dedekind. 
Remind that an infinite non-abelian group, satisfying the minimal condition for abelian non-
normal subgroups, is called an I-group (S.N. Chernikov [5]). Let, as usual, min-ab be the minimal 
condition for abelian subgroups. S.N.Chernikov has disclosed the structure of non-periodic I-groups 
and has established the important properties of periodic I-groups without min-ab (see in [5] his 
Theorems 4.6, 4.10, 4.11). The following new theorem gives the description of periodic I-groups 
without min-ab. Below pA is a Sylow p-subgroup of the group A and e(A) is its exponent. 
Theorem 2 (N.S. Chernikov). For the periodic group G the following statements are equivalent. 
(i) G is an I-group without min-ab. 
Известия Гомельского государственного университета 
имени Ф. Скорины, № 3 (84), 2014 
N.S. Chernikov 
 
122 
(ii) Either G is Hamiltonian non-Chernikov, or for some b∈G and some Dedekind non-
Chernikov subgroup GA with Chernikov 2A and with ∞<=< AGm :1 , G=A and 
)(AZbA >⊆< and b induces on any pA an automorphism of certain order mnp | , raising each 
element of pA to some power, and also: 1) 12 =n , if A is Hamiltonian; 2) for every >∈< bt , every 
finite subgroup AF ⊆ and any Ff ∈ with )(Fef =>< , and for an integer r with rt ff = and 
every ,Fa∈ rt aa = and )()( FCfC tt ><>< = ; 3) 
k
p spn = with some natural 1| −ps and integer 
k>0; in particular, 11 −<< pnp if pnp |/ , and 0,22 >= kn
k ; 4) if pnp | and 2≠p , then pA is finite; 
5) if 2A is infinite and [ ] 1,2 ≠bA , then for any 2Aa∈ , 1−= aab ; 6) if pnp |/ and for >∈< bt 
[ ] 1, ≠tAp , then for every },{\Aa p 1∈ aat ≠ ; 7) the subgroup pp npA |:1,< and >≠ 2p (respec-
tively >≠< mnA pp :1, ) is finite (respectively Chernikov). 
Now let nabmin− be the minimal condition for non-abelian non-normal subgroups. The in-
finite non-abelian groups with normal infinite non-abelian subgroups present some special case of 
groups with nabmin− . These infinite groups, are called IH -groups (S.N.Chernikov, see, for in-
stance, [5]). S.N.Chernikov has obtained a series of principal results relating to the IH -groups (see, 
for instance, [5, Chapter 6] and [6, §§3-5]). N.S.Chernikov (see, for instance, Theorem 3.2 [6]) 
gives an affirmative answer to the following S.N.Chernikov’s question (see [7, P.20]): Are the bina-
ry graded IH -groups solvable? 
The following two principal propositions hold. 
Theorem 3 (N.S.Chernikov [3, Theorem 3]). The non-abelian group G∈Ł satisfies the condi-
tion nabmin− iff it is a Chernikov group or a solvable group with normal non-abelian subgroups. 
Proposition 1 (N.S.Chernikov). The IH -group G is solvable iff G∈Ł. 
The following result is an immediate consequence of Proposition 1. 
Corollary 2 (N.S.Chernikov, see Theorem 3.2[6]). The binary graded IH -groups are solvable. 
Proof of Theorem 1. Let G be non-Artinian. It is easy to see: G contains some non-Artinian 
subgroup H such that any its non-Artinian subgroup is normal in G; H has some descending series 
1210 ⊇=⊃⊃⊃⊃=
<
)H(H...HHHH 
γα
αγ such that γH is Artinian. Obviously all 
γαα <,H/H , are Dedekind. So by R.Baer’s Theorem [1], γαα <≤ ,)'H/H( 2 . Therefore 
2≤)'H/H( γ . Put γγ H/L)'H/H( = . Then L is Artinian and H /L is abelian non-Artinian. Since 
G /L does not satisfy min-ab and satisfies nmin− , it is Dedekind (see Corollary 2[2]). Put (G /L)′ 
=K/L. Then G′⊆K and |K /L|≤2. So K and G′ are Artinian. Suppose that G′∈Ł. The class Ł coincides 
with the minimal local class of groups, closed with respect to forming series and containing the 
class of weakly graded groups (see Proof of Theorem 3[3]). Consequently the class Ł coincides 
with the minimal local class of groups, closed with respect to forming series, subcartesian products 
and containing the class of weakly graded groups. So by Theorem C (iii) [8], G′ is Chernikov. In 
consequence of S.N.Chernikov’s Theorem [2], G is Chernikov or Dedekind. 
Proof of Corollary 1. If G satisfies nmin− , then by Theorem 1, G′ is Artinian and so G′is 
periodic weakly graded and G is Chernikov or Dedekind. 
Proof of Theorem 2. Let (i) hold and A be the subgroup of G generated by all its abelian non-
Artinian subgroups. Clearly A is non-Chernikov. In view of S.N.Chernikov’s Theorem 4.11[5], A is 
Dedekind, all its subgroups are normal in G, and for some g∈G, G=A<g>. If G=A, then G is Hamil-
tonian non-Chernikov. Let G≠A and K={a∈A: |<a>|≤2}. Since >< a G, a∈Z(G). Therefore K is an 
elementary abelian central 2-subgroup of G. Since <g>K⊄A and <g>K is abelian, K is Artinian 
and so finite. Therefore 2A is Chernikov. 
Groups with some minimal conditions for non-normal subgroups 
 
123 
Put b=g if A is abelian. Let A be Hamiltonian. Then by R.Baer’s Theorem [1], A=Q×E×R 
with Q, E and R as above. Clearly R⊆Z(A), E⊆K(⊆Z(G)) and 2A =Q×E, A= 2A ×R. If [g, 2A ]=1, then 
obviously A∩<g>⊆Z(A). Put again b=g. Let now [g, 2A ]≠1. For Sylow 2- and 2′-subgroups <h> 
and <c> of <g> resp., <g>=<h>×<c>. For distinct subgroups , <v>⊆Q of order 4, 
Q=<u,v>. Since Gv,u ><>< and 2 |/ |<c>|, [c,u]=[c,v]=1. Then, because of E⊆Z(G), [c, 2A ]=1 
and [h, 2A ]=[h,Q]≠1. Clearly, we may assume that [h,u]≠1 and [h,v]≠1. Then 
1−= uuh and 1−= vvh . 
Consequently uuhuv = and vvhuv = , and [huv, 2A ]=1. Since Guv >< , huv is a 2-element. Clearly 
[huv,c]=1. Put now b=(huv)c. Then G=A. Since [huv, 2A ]=[c, 2A ]=1, we have [b, 2A ]=1 and, 
obviously, A∩⊆Z(A). The assertion 1 of the statement (ii) is correct. 
Consider the assertion 2 of (ii). Show that rt aa = . First, let F be abelian. For some sub-
group D, F=<f>×D. Since <f>, <fd>, GD , for some integer s: ttssst dfdf)fd()fd( === ; so 
Ddf strs ∈= −− and 1=−rsf . Since obviously <fd>∩D=1, ( )s r s r s rfd f d− − −= = 
= 1s rd fd D− ∈< > ∩ = . 
Thus rrrsrrsst )fd()fd()fd()fd()fd()fd( ==== −+− . Since F=<fd:d∈D>, for every a∈F, 
rt aa = . 
Let F be Hamiltonian. Then for some abelian subgroup *F of F, <a> 'F2 ⊆
*F and |F: *F |=2. 
Take *f ∈ *F with |< *f >|=e( *F ). It is easy to see: < *f >∼<f>. Let >=<>< 22 ff , 
>=<>< '' ff 22 , >=<><
** ff 22 , >=<><
*
''
* ff 22 . In view of the assertion 1, 22 ff
t = and 
*t* ff 22 = . Since <f>
'F2 is abelian, 
r
'
t
' ff 22 = and 
r*
'
t*
' ff 22 = (see above). Consequently, because 
of rt ff = , we have r*t* ff = . Since *F is abelian and a∈ *F , rt aa = (see above). 
 Let w∈ )f(C t>< . Since 
wf =f, as above for every a∈F, wa =a. Thus )f(C t>< = )F(C t>< . 
Consider the assertion 3 of (ii). Let pn ≠1. For each w∈\ )A(C pb>< , there exists wa ∈ pA such 
that [w, wa ]≠1. Let F be the subgroup generated by all wa . Then F is finite and 
)F(C b>< = )A(C pb>< . In view of the assertion 2, for some f∈F, )F(C b>< = )f(C b ><>< . So b induc-
es on <f> an automorphism of order pn . Thus pn |Aut<f>|. But |Aut<f>|=(p-1)
sp , s>0. 
 Consider the assertions 4 and 5 of (ii). Let p≠2 and p pn , or p=2 and [ 2A ,t]≠1 with t=b. In 
the first case, for some p-element t∈, [ pA ,t]≠1. Put K={u∈ pA : ||<p}. Since uG, K<t> is 
clearly an abelian p-subgroup. Since clearly K<t>⊄A, K<t> is Artinian. So K is finite. Then pA is 
Chernikov (see, for instance, Theorem 7.9 [6]). In the second case, pA is also Chernikov. 
 Let pA be infinite. Then it contains some quasicyclic subgroup N. Let a,d∈ pA and also 
[d,t]≠1. Let f∈N and also |<a>|, |<d>| < |<f>|. Then e(<a><d><f>)=|<f>|. In view of the assertion 
2, ff t ≠ . Since [N,t]≠1 and NG, N<t> is not a p-subgroup with odd p. Then p=2 and 1−= ff b . 
In view of the assertion 2, 1−= aab . 
 Consider the assertion 6. Note that p≠2: otherwise kpn 2= , k>0, (see the assertion 3) and 
p pn which is a contradiction. It is enough to consider the case when |<a>|=p. For some d∈ pA , 
≠td d. Let c∈<d> and |<c>=p. Since pnp |/ , )d(C:t|p t ><></ >< . Consequently for some natural 
r such that 1<r<p, rt cc = . By virtue of the assertion 2, 1≠= rt aa . 
 Consider the assertion 7. In view of the assertion 4 | pA |<∞, if p pn and p≠2. Since pn n, the 
set of all p, for which p pn , is finite or empty. Thus the subgroup < pA ,1: p pn and p≠2> is finite. 
Further, assume that < pA ,1: pn ≠m> is not Chernikov. Then by Kurosh’es Theorem (see, for instance, 
Proposition 4.2.11 [9]), it is not Artinian. If pn ≠m, then for some w∈\A, [w, pA ]=1. Since <w> pA 
is abelian and <w> pA ⊄A, <w> pA is Artinian. It is easy to see: there exist some infinite set π of primes 
and some w∈\A such that for each p∈π, pA ≠1 and <w> pA is abelian. Then <w>( pp Aπ∈× ) is abeli-
an non-Artinian and also <w>( pp Aπ∈× )⊄A, which is a contradiction. 
N.S. Chernikov 
 
124 
 Now let (ii) hold. Obviously G is non-abelian and contains some Dedekind non-Chernikov 
subgroup. In view of R.Baer’s Theorem [1], this subgroup has an abelian subgroup of finite index. 
By virtue of Kurosh’es Theorem, the last is non-Artinian. Thus G does not satisfy min-ab. So if G is 
Hamiltonian non-Chernikov, then (i) holds. Assume that G is not such group. Let U be its abelian 
subgroup. If U⊆A, then obviously U G. Let U⊄A and v∈U\A. Then v=dt with d∈A and t∈\A. By 
virtue of the assertion 7, the subgroup L= 2A < pA ,1: p pn ><< pA ,1: pn ≠m> is Chernikov. Fur-
ther, for every odd p such that pnp |/ and pn =m, and for every a∈ pA , aa
t ≠ if a≠1 (see the asser-
tion 6). So aaaa tdtv ≠== . Consequently, U∩ pA =1. Therefore U∩A=U∩L. So U∩A is Cher-
nikov. Then U is Chernikov. Thus (ii) necessitates (i). 
 Proof of Proposition 1. Indeed, let G∈Ł . Since G obviously satisfies nabmin− , it is almost 
solvable (see Theorem 3). Then in consequence of Proposition 6.5 [5], it is solvable. 
 In conclusion, mention the following interesting result arising from Theorems 4.11, 4.10 [5]. 
 Proposition 2 (N.S.Chernikov). A group satisfies the minimal condition for abelian non-
normal subgroups iff every its abelian subgroup is Chernikov or normal. 
 
References 
 
 1. Baer, R. Situation der Untergruppen und Structur der Gruppe/ R. Baer // S.-B. Heidelberg. Akad. 
Math. – Nat. Klasse. – 1933. – Bd. 2. – S. 12–17. 
 2. Chernikov, S.N. The infinite non-abelian groups with the minimal condition for non-normal sub-
groups / S.N. Chernikov // Mat. Zametki. – 1969. – Vol. 6, № 1. – P. 11–18. (in Russian) 
 3. Chernikov, N.S. Groups satisfying the minimal condition for non-abelian non-normal subgroups / 
N.S. Chernikov // J.Sib. Federal Univ. Math.&Physics. – 2014. – Vol. 7, № 1. – P. 22–34. 
 4. Ol’shanskiy, A.Yu. Geometry of defining relations in groups/ A.Yu.Ol’shanskiy. – Moskva : Nau-
ka, 1989. – 442 p. (in Russian) 
 5. Chernikov, S.N. Groups with prescribed properties of the system of subgroups / S.N. Chernikov. – 
Moskva : Nauka, 1980. – 384 p. (in Russian) 
 6. Chernikov, N.S. S.N.Chernikov and the theory of groups. The Kiev period / N.S.Chernikov/ In-te 
Math. Nat. Acad. Sci., Ukraine, Kyiv, 2012. – 106 p. (Preprint / NAS of Ukraine. In-te math.; 2012.01.). 
(in Russian) 
 7. Chernikov, S.N. Infinite groups defined by properties of the system of infinite subgroups / 
S.N.Chernikov// Sixth All-Union Symposium on Group Theory (Cherkassy, 1978). – Naukova dumka, Kyiv, 
1980. – P. 5–22. (in Russian) 
8. Chernikov, N.S. Groups with minimal conditions/ N.S.Chernikov// Fundament. i prykl. matemat-
ika. – 2008. – Vol. 14, № 5. – P. 219–235. (in Russian). Engl. transl. : J. Math. Sci. – 2009. – Vol. 163, № 6. 
– P. 774–784. 
9. Robinson, D.J.S. A course in the theory of groups/ D.J.S. Robinson // Springer : New York etc., 
1980. – 482 p. 
 
 
Институт математики 
Национальной академии наук Украины Поступила в редакцию 14.05.2014

Groups with some minimal conditions for non-normal subgroups

Abstract

Similar works

Full text

Available Versions

Repository of Gomel State University