3МАТЕМАТИКА
Ц и т у в а н н я: Kurdachenko L.A., Pypka A.A., Subbotin I.Ya. On groups, whose non-normal subgroups are 
either contranormal or core-free. Допов. Нац. акад. наук Укр. 2020. № 10. С. 3—8. https://doi.org/10.15407/
dopovidi2020.10.003
ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2020. № 10: 3—8
https://doi.org/10.15407/dopovidi2020.10.003
UDC 512.544
L.A. Kurdachenko 1, A.A. Pypka 1, I.Ya. Subbotin 2
1 Oles Honchar Dnipro National University, Ukraine
2 National University, Los Angeles, USA
E-mail: lkurdachenko@i.ua, pypka@ua.fm, isubboti@nu.edu
On groups, whose non-normal subgroups 
are either contranormal or core-free
Presented by Corresponding Member of the NAS of Ukraine V.P. Motornyi
We investigate the influence of some natural types of subgroups on the structure of groups. A subgroup H of a group 
G is called contranormal in G, if G = HG. A subgroup H of a group G is called core-free in G, if CoreG(H) = 〈1〉. 
We study the groups, in which every non-normal subgroup is either contranormal or core-free. In particular, we obtain 
the structure of some monolithic and non-monolithic groups with this property.
Keywords: normal subgroup, contranormal subgroup, core-free subgroup, monolithic group.
ОПОВІДІ
НАЦІОНАЛЬНОЇ 
АКАДЕМІЇ  НАУК
УКРАЇНИ
Let G be a group. The following two normal subgroups are associated with any subgroup H of 
group G: HG, the normal closure of H in a group G, the least normal subgroup of G including H, 
and CoreG(H), the normal core of H in G, the greatest normal subgroup of G which is contained 
in a subgroup H. We have
|G xH H x G= 〈 ∈ 〉
and
Core ( ) xG
x G
H H
∈
=  .
A subgroup H is normal, if and only if H = HG = CoreG(H). In this sense, the subgroups H, 
for which CoreG(H) = 〈1〉, are the complete opposites to normal subgroups. A subgroup H of a 
group G is called core-free in G, if CoreG(H) = 〈1〉.
A subgroup H is normal, if and only if H = HG. In this sense, the subgroups H, for which 
HG = G, are the complete opposites to the normal subgroups. A subgroup H of a group G is 
called contranormal in G, if G = HG. J.S. Rose has introduced the term “a contranormal subgroup” 
in paper [1].
4 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2020. № 10
L.A. Kurdachenko, A.A. Pypka, I.Ya. Subbotin
For each subgroup H of a group G, we have the following two extreme and opposite si-
tuations:
HG = H or HG = G,
and, respectively,
CoreG(H) = H or CoreG(H) = 〈1〉.
The following extreme cases immediately appear.
The first case: every proper subgroup of G is normal. Such group is called a Dedekind group. 
A Dedekind group G has the following structure: it is either Abelian, or G = Q8 × D × P where 
Q8 is a quaternion group of order 8, D is an elementary Abelian 2-group and P is an Abelian 
2′-group [2].
The second case: every proper subgroup of G is core-free. In this case, group G does not in-
clude proper non-trivial normal subgroups, that is, G is a simple group. However, a simple group 
include normal subgroups: they are G and the trivial subgroup.
The third case: every proper non-trivial subgroup of G is contranormal. In this case, group 
G does not include proper non-trivial normal subgroups, so that again G is a simple group.
In the last two cases, we came only to simple groups. Note again that a simple group has 
the only three following types of subgroups: normal, core-free, and contranormal.
Therefore, the following question naturally appears: what can we say about the groups, 
whose subgroups are either normal, core-free or contranormal?
In the process of this study, the need of a separate study of the following two types of 
groups naturally arises: the groups, whose subgroups are either normal or core-free, and the 
groups, whose subgroups are either normal or contranormal.
Note that groups having only two types of subgroups, which are also antagonistic in some 
sense to each other, have been considered by many authors. Here, we provide the citation on 
some of them, whose subjects are to some extent related to this topic [3-13].
Groups, whose subgroups are either normal or core-free, have been studied in [14, 15].
The study of groups, whose subgroups are either normal or contranormal was initiated in 
paper [5]. In Theorem 2 of [5], basis structural features of such groups were shown.
A group G is called quasisimple, if a central factor-group G/ζ (G) is simple and G = [G,G].
Let G be a quasisimple group, and let H be a non-trivial subgroup of G. If ζ (G) does not 
include H, then Hζ (G)/ζ (G) is non-trivial. Since G/ζ (G) is simple, HGζ (G)/ζ (G) = G/ζ (G), 
that is HGζ (G) = G. If we suppose that HG ≠ G, then G/HG is Abelian, what is impossible. Hence, 
HG = G. Thus, every subgroup of a quasisimple group is either normal or contranormal.
Theorem A. Let G be a group, whose non-normal subgroups are contranormal. If G is not so-
luble, then G is simple or quasisimple group.
Let D be a Prüfer 2-group, 2 21 1| 1, ,n n nD d d d d n+= 〈 = = ∈ 〉 , and Q be a quaternion group 
of order 8, Q = 〈 u1 〉 〈 u2 〉 where | u1 | = | u2 | = 4,
〈 c 〉 = 〈 u1 〉 ∩ 〈 u2 〉
and c = [u1, u2]. Consider the semidirect product H = D  Q, where CQ(D) = 〈 u1 〉 and 2 1ud d −=  
for all elements d ∈ D. Put 12 1v d u−= , then
5ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2020. № 10
On groups, whose non-normal subgroups are either contranormal or core-free
2 22 2 21 1 1 1 1 1 1 1 1 1 1
2 1 1 2 2 1 2 1 2 12 1( ) ( ) ( ) ( ) ( )
u uu u uv d u d u d u d u d u d u v− − − − − − − − − − −= = = = = = = .
It shows that the subgroup V = 〈 v 〉 is normal in H. Put Q∞ = H/V, A = DV/V, y = u2V, then 
A ≅ D is clearly a normal Prüfer 2-subgroup of Q∞, | y | = | u2 | = 4. We have d2V = u1V, so that
2 2 2 2 2
2 2 1 2( )y u V u V u V d V A= = = = ∈ .
It follows that 〈 y 〉 ∩ A = Ω1(A), and ay = a —1 for all elements a ∈ A. The constructed 
above group Q∞ is called the infinite generalized quaternion group.
Theorem B. Let G be a soluble group, whose non-normal subgroups are contranormal. Suppose 
that G is not a Dedekind group.
If G is a p-group for some prime p, then p = 2, and G is a group of one of the following types of groups:
(i) G = D  〈 g 〉, where D is a normal divisible Abelian 2-subgroup, element g has order 2 or 
4, d  g = d –1 for every element d ∈ D;
(ii) G = D 〈 g 〉, where D is a normal divisible Abelian 2-subgroup, element g has order 2 or 4, 
d g = d –1 for every element d ∈ D, D = A × B, where A is a Prüfer 2-subgroup, g2 ∈ Ω1(A), and 〈 A,g 〉 
is an infinite generalized quaternion group.
If G is a periodic group and | Π(G) | . 2, then G is a group of one of the following types:
(iii) G = S  〈 g 〉, where g is a p-element for some prime p, S is an Abelian Sylow p′-subgroup 
of G, C〈 g 〉(S) = 〈 g p 〉, and every subgroup of S is G-invariant;
(iv) G = D  〈 g 〉, where D is a normal Abelian subgroup, D = S × K, where S is a Sylow 
2′-subgroup of G, K is a divisible 2-subgroup, element g has order 2 or 4, d g = d –1 for every ele-
ment d ∈ D;
(v) G = D 〈 g 〉, where D is a normal Abelian subgroup, D = S × A × B, where S is a Sylow 2′-
subgroup of G, A × B is a divisible 2-subgroup, element g has order 2 or 4, d g = d –1 for every element 
d ∈ D, g2 ∈ Ω1(A), and 〈 A,g 〉 is an infinite generalized quaternion group.
If G is a non-periodic group, then G is a group of one of following types:
(vi) G = D 〈 g 〉, an element g has order 2 or 4, and xg = x–1 for each element x ∈ D, 
C〈 g 〉(D) = 〈 g2 〉, 2 ∉ Π(D), D2 = D, and every subgroup of D is G-invariant;
(vii) G = D  〈 g 〉, an element g has order 2 or 4, and xg = x–1 for each element x ∈ D, 
C〈 g 〉(D) = 〈 g2 〉, D2 = D, D = S × B, 2 ∉ Π(B), S is a divisible Sylow 2-subgroup of D;
(viii) G = D 〈 g 〉, an element g has order 2 or 4, and xg = x–1 for each element x ∈ D, 
C〈 g 〉(D) = 〈 g2 〉, D 2 = D, D = S × B, where 2 ∉ Π(B), S = A × C is a divisible Sylow 2-subgroup of 
D, 〈 g 〉 ∩ (A × B) = 〈1〉, g 2 ∈ Ω1(C) and 〈 C,g 〉 is an infinite generalized quaternion group.
Further, at the study of groups, whose subgroups either are normal, core-free or contranor-
mal, it is natural to assume that they include proper contranormal and proper core-free sub-
groups. In this paper, we considered such groups imposing an additional restriction that they are 
periodic and locally soluble. Their description decomposes to few natural parts.
Theorem C. Let G be a group, whose subgroups are either normal or contranormal or core-free. 
If G is locally soluble, then G is a soluble group.
Let G be a group and A be a normal subgroup of G. The intersection MonG(A) of all non-
trivial G-invariant subgroups of A is called the G-monolith of A. If MonG(A) is not trivial, then 
subgroup A is called G-monolithic. If A = G, then we will say that A is the monolith of group G 
and denote it by Mon(G).
6 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2020. № 10
L.A. Kurdachenko, A.A. Pypka, I.Ya. Subbotin
The following theorem is devoted to the monolithic case.
Theorem D. Let G be a soluble periodic monolithic group, whose non-normal subgroups are 
contranormal or core-free. Suppose that G includes proper contranormal and core-free subgroups. 
Then G is a group of one of following types:
(i) G = D  〈 v 〉, where D is a normal Prüfer 2-subgroup, v2 = 1, d v = d –1 for all elements d ∈ D;
(ii) G = M  S, M is an elementary Abelian p-subgroup, p is a prime, S is a locally cyclic p′-
subgroup, CG(M) = M, and every complement to M in G is conjugate with S. In particular, if M is fi-
nite, then G is finite, and G = M  S, where S is a cyclic Sylow p′-subgroup of G;
(iii) G = D  〈 g 〉, where D is a normal cyclic p-subgroup, | g | = q is a prime, q < p, CG(D) = D;
(iv) G = D  〈 g 〉, where D is a normal Prüfer p-subgroup, | g | = q is a prime, q < p, 
CG(D) = D;
(v) G = D  〈 g 〉, where D is an extraspecial p-subgroup, p is a prime, | g | = q is a prime, q < p, 
q ≠ 2, moreover, M = [D,D] = ζ(D) is a monolith of G, and every subgroup of D/M is G-invariant;
(vi) G = M  K, where M is a finite elementary Abelian p-subgroup, p is an odd prime, K is 
a quaternion group of order 8, M is a minimal normal subgroup of G, CG (M) = M;
(vii) G = M  B, where M is a minimal normal elementary Abelian p-subgroup, p is an odd 
pri me, B = K  〈 u 〉 where K is a normal Prüfer 2-subgroup, u2 = 1, au = a –1 for each a ∈ K;
(viii) G = M  B, where M is a minimal normal elementary Abelian p-subgroup, p is an odd 
prime, B is an infinite generalized quaternion group;
(ix) G = M  V, where M is a minimal normal elementary Abelian p-subgroup, p is a prime, 
V = D1  〈 g 〉 where D1 is a locally cyclic p′-subgroup, | g | = p, every subgroup of D1 is 〈 g 〉-in-
variant, CV(D1) = D1;
(x) G = M  V, where M is a minimal normal elementary Abelian p-subgroup, p is a prime, 
V = D1  〈 g 〉, where D1 is a locally cyclic subgroup, g is a q-element, q is an odd prime, p, q ∉ Π(D1), 
C〈 g 〉 (D1) = 〈 gq 〉, and every subgroup of D1 is 〈 g 〉-invariant;
(xi) G = M  V, where M is a minimal normal elementary Abelian p-subgroup, p is a prime, 
V = D1  〈 g 〉, where D1 is a locally cyclic subgroup, g is a 2-element, 2, p ∉ Π(D1), C〈 g 〉(D1) = 〈 g2 〉, 
xg = x –1 for each element x ∈ D1;
(xii) G = M  V, where M is a minimal normal elementary Abelian p-subgroup, p is a prime, 
V = (S × K)  〈 g 〉, where S × K is a locally cyclic subgroup, moreover, K is a Prüfer 2-subgroup, S 
is a 2′-subgroup, | g | = 2, xg = x–1 for each element x ∈ S × K;
(xiii) G = M  V, where M is a minimal normal elementary Abelian p-subgroup, p is a prime, 
V = S  (K〈 g 〉), where S is a locally cyclic 2′-subgroup, K is a Prüfer 2-subgroup, | g | = 4, g2 ∈ Ω1(K), 〈 K,g 〉 is an infinite generalized quaternion group, CV(S) = S × K, xg = x –1 for each element 
x ∈ S × K.
Finally, the last theorem considers the non-monolithic case.
Theorem E. Let G be a soluble periodic non-monolithic group, whose non-normal subgroups 
are either contranormal or core-free. Suppose that G includes proper contranormal and core-free 
subgroups. Then G is a group of one of the following types:
(i) G = A  〈 v 〉, where A is a normal divisible 2-subgroup, v2 = 1, av = a–1 for all elements 
a ∈ A;
(ii) G = M  (〈 c 〉 × 〈 g 〉), where M is a normal subgroup, having prime order p ≠ 2, | c | = s is 
a prime, | g | = q is a prime, q ≠ s, q divides p — 1, CG(M) = M × 〈 c 〉;
7ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2020. № 10
On groups, whose non-normal subgroups are either contranormal or core-free
(iii) G = M  (〈 c 〉 × 〈 g 〉), where M is a normal subgroup of prime order p ≠ 2, | c | = | g | = q 
is a prime, q divides p — 1, CG (M) = M × 〈 c 〉;
(iv) G = M  〈 g 〉, where M is a normal subgroup of prime order p ≠ 2, g is an element of order 
q, q is a prime, q divides p — 1, CG (M) = M;
(v) G = [G,G]  〈 g 〉, where [G,G] is a normal cyclic p-subgroup, where p is an odd prime, 〈 g 〉 
is a cyclic q-subgroup, q is a prime, CG([G,G]) = [G,G];
(vi) G = [G,G]  〈 g 〉, where [G,G] is a normal Prüfer p-subgroup, where p is an odd prime, 
〈 g 〉 is a cyclic q-subgroup, q is a prime, CG ([G,G]) = [G,G];
(vii) G = (〈 a1 〉 × 〈 a2 〉)  〈 g 〉, where | a1 | = | a2 | = q, | g | = p, p is a prime, p < q, CG ([G,G]) = [G,G], 
11
g ma a= , 22
g sa a= , 1 - m, s < q, m ≠ s;
(viii) G = [G,G]  〈 g 〉, where g is an element of order p, p is a prime, p < q, [G,G] is an Abelian 
Sylow q-subgroup of G, CG([G,G]) = [G,G], and every subgroup of [G,G] is G-invariant;
(ix) G = S  〈 g 〉, where g is an element of order 2, S is an Abelian 2′-subgroup, CG (S) = S, 
and xg = x –1 for every element x ∈ S;
(x) G = S  P, where P is a Sylow 2-subgroup of G and S is an Abelian 2′-subgroup, P = P1  〈 g 〉, 
where D is a normal divisible Abelian 2-subgroup, [S,P1] = 〈1〉, | g | = 2, and xg = x –1 for every ele-
ment x ∈ S × P1;
(xi) G = S  〈 g 〉, where | g | = p, where p is the least prime of the set Π(G), S is an Abelian 
Sylow p′-subgroup of G, CG(S) = S, and every subgroup of S is G-invariant.
The first two authors supported by the National Research Foundation of Ukraine (Grant 
No. 2020.02/0066).
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L.A. Kurdachenko, A.A. Pypka, I.Ya. Subbotin
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Received 24.12.2019
Л.А. Курдаченко 1, 
О.О. Пипка 1, І.Я. Субботін 2
1 Дніпровський національний університет ім. Олеся Гончара
2 Національний університет, Лос-Анджелес, США
E-mail: lkurdachenko@i.ua, pypka@ua.fm, isubboti@nu.edu 
ПРО ГРУПИ, ПІДГРУПИ ЯКИХ АБО НОРМАЛЬНІ, 
АБО КОНТРАНОРМАЛЬНІ, АБО ВІЛЬНІ ВІД ЯДРА
Досліджено вплив деяких типів підгруп на структуру груп. Підгрупу H групи G називаємо контранор-
маль ною в G, якщо G = HG. Підгрупу H групи G називаємо вільною від ядра в G, якщо CoreG(H) = 〈1〉. 
Вивчено групи, в яких кожна підгрупа або нормальна, або контранормальна, або вільна від ядра. Точніше, 
одержано будову деяких монолітичних та немонолітичних груп з цією властивістю”.
Ключові слова: нормальна підгрупа, контранормальна підгрупа, вільна від ядра підгрупа, монолітична група.