CC BY
12
ЧЕБЫШЕВСКИЙ СБОРНИК
Том 20. Выпуск 2.
УДК 512.542 DOI 10.22405/2226-8383-2019-20-2-391-398
О дисперсивных по Ope F-гпперцентральных подгруппах
конечных групп
В. И. Мурашко
Мурашко Вячеслав Игоревич — магистрант, Гомельский государственный университет им. Ф. Скорины (г. Гомель, Республика Беларусь). e-mail: mvimath@yandex.ru
Аннотация
Рассматриваются только конечные группы. Пусть А — группа автоморфмизмов группы G, содержащая все внутренние автоморфизмы, и F — максимальный внутренний локальных экран насыщенной формации F- А-композиционный фактор Н/К группы G называется A-F-центрадьным, если А/Са(Н/К) G F(р) для всех р G -к(Н/К). A-F-гиперцентром G называется наибольшая А-допустимая подгруппа G, все А-композиционные факторы ниже которой А-F-центрадьны. Обозначается Z^(G,A).
Напомним, что группа G называется дисперсивной по Ope, если G имеет нормальную холлову {р1,... ,^}-подгруппу для 1 < г < п, где р1 > ■ ■ ■ > рп — все простые делители |G|. Главным результатом работы является: Пусть F — наследственная насыщенная формация, F — её максимальный внутренний локальный экран и N — дисперсивная по Ope А-допустимад подгруппа группы G, где InnG < А < AutG. Тогда и только тогда N < Z$(G,A), тогда Na(P)/Са(Р) G F(р) для любых силовской ^подгруппы Р группы N и простого делителя р порядка N.
В качестве следствий были получены известные результаты Р. Бэра о нормальных подгруппах в сверхразрешимом гиперцентре и элементах гиперцентра.
Пусть G — группа. Напомним, что
Ln (G) = {х g G | [x, ai,..., an] = 1 у ai,... ,an g AutG}
и G называется автонпльпотентной, если G = Ln(G) для некоторого натурального п. Из главного результата можно извлечь критерии автонильпотентности групп. В частности, группа G автонильпотентна тогда и только тогда, когда она является прямым произведением своих силовских подгрупп и группа автоморфизмов любой силовской р-подгруппы группы G является ^группой для любого простого делителя р порядка G. Приведены примеры автонильпотентных групп нечетного порядка.
Ключевые слова: Конечная группа, нильпотентная группа, сверхразрешимая группа, автонильпотентная группа, A-F-гиперцептр группы, наследственная насыщенная формация.
Библиография: 25 названий. Для цитирования:
F
Чебышевский сборник, 2019, т. 20, вып. 2, с. 391-398.
CHEBYSHEVSKII SBORNIK Vol. 20. No. 2.
UDC 512.542 DOI 10.22405/2226-8383-2019-20-2-391-398
On the F-hypercentral subgroups with the sylow tower property of
finite groups
V. I. Murashka
Murashka Viachaslau Igaravich — graduate student, Francisk Skorina Gomel State University (Gomel, Republic of Belarus). e-mail: mvimath@yandex.ru
Abstract
Throughout this paper all groups are finite. Let A be a group of automorphisms of a group G that contains all inner automorphisms of G and F be the canonical local definition of a saturated formation F- An A-composition factor H/K of G is railed A-F-central if A/Ca(H/K) G F(p) for all p G -k(H/K). The A-F-hypercenter of G is the largest A-admissible subgroup of G such that all its A-composition factors are A-F-central. Denoted by Z$(G,A).
Recall that a group G satisfies the Sylow tower property if G has a normal Hall {pi,... ,Pi}-subgroup for all 1 < i < n where pi > ■ ■ ■ > pn are all prime divisors of |G|. The main result of this paper is: Let F be a hereditary saturated formation, F be its canonical local definition and N be an A-admissible subgroup of a group G where InnG < A < AutG that satisfies the Sylow tower property. Then N < Z%(G, A) if and only if NA(P)/CA(P) g F(p) for all Sylow ^subgroups P of N and every prime divisor p of iN
As corollaries we obtained well known results of R. Baer about normal subgroups in the supersoluble hypercenter and elements in the hypercenter.
Let G be a group. Recall that Ln(G) = {x G G | [x, a1,..., an] = 1 V«i,... ,an G AutG} and G is autonilpotent if G = Ln(G) for some natural n. The criteria of autonilpotency
of a group also follow from the main result. In particular, a group G is autonilpotent if and only if it is the direct product of its Sylow subgroups and the automorphism group of a Sylow p-subgroup of G is a p-group for all prime divisors p of |G|. Examples of odd order autonilpotent groups were given.
Keywords: Finite group, nilpotent group, supersoluble group, autonilpotent group, hypercenter of a group, hereditary saturated formation.
Bibliography: 25 titles. For citation:
V. I. Murashka, 2019, "On the F-hvpercentral subgroups with the svlow tower property of finite groups" , Chebyshevskii sbornik, vol. 20, no. 2, pp. 391-398.
1. Introduction and results
Throughout this paper all groups are finite and G always denotes a finite group. Recall that AutG and InnG are the groups of all and inner automorphisms of G respectively.
Let A be a group of automorphisms of G. Kaloujnine [1] and Hall [2] showed that if A stabilizes some chain of subgroups of G, then A is nilpotent. Huppert [3] and Shemetkov [4] showed that if G has has A-admissible series with prime indexes, then A is supersoluble. Shemetkov [4] and Schmid
FF
G with respect to A played an important role in their research.
Recall that ■k(G) is the set of all prime divisors of |G|. A formation is a class F of groups with the following properties: (a) every homomorphic image of an F-group is an F-group, and (b) if G/M and G/N are F-groups, then G/(M n N) G F- A formation F is said to be: saturated if G G F whenever G/§(G) G F where $(G) is the Frattini's subgroup of G\ hereditary if H G F whenever H < G G F- A function of the form f : P ^ {formations} is called a formation function. Recall [6, p. 356] that a formation F is called local if F = (G | G/Cg(H/K) G f (p) for every p G n(H/K) and every chief factor H/K of G) for some formation function f. In this case f is called a local F
F
function F, defini ng F such th at F (p) = NPF (p) C F for every p G P by Proposition 3.8 [6, p. 360]. In this case F is called the canonical local definition of F-
Let InnG < A be a group of automorphisms of G and F be the canonical local definition of a local formation F- An ^-composition factor H/K of G is called A-F-central if A/Ga(H/K) G F(p) for all p G n(H/K). The A-F-hypercenter of G is the largest A-admissible subgroup of G such that all its A-composition factors are A-F-central. This subgroup always exists by Lemma 6.4 [6, p. 387]. It is denoted by Z$(G, A). If A = InnG, then it is just the F-hvpercenter Z$(G) of G. If F = N is the class of all nilpotent groups, then we use Z^(G, A) to denote the A-hvpercenter ZN(G,A) of G. Recently the subgroups of Z^(G,A) have been studied for example in [7, 8, 9, 10, 11].
Recall that Sy\pG is the set of all Svlow subgroups of G\ G satisfies the Sylow tower property if G has a normal Hall {p\,...,pj}-subgroup for all 1 < i < n where p\ > ■ ■ ■ > pn are all prime divisors of |G|. It is well known that a supersoluble group satisfies the Svlow tower property. Recently series of hereditary saturated formations of groups that satisfy the Sylow tower property have been constructed (see, [12, 13, 14, 15]).
Theorem 1. Let, F be a hereditary saturated formation, F be its canonical local definition and N be an A-admissible subgroup of G where InnG < A < AutG that satisfies the Sylow tower property. Then N < Zf(G, A) if and only if NA(P)/CA(P) G F(p) for all P G Sy\p(N) and p G n(N).
Author obtained particular cases of this theorem for A = InnG and two formations of supersoluble type in [16, 17]. Recall that a group G is called strictly p-closed if G/Op(G) is abelian of exponent dividing p — 1. We use U to denote the class of all supersoluble groups.
Corollary 1 (R. Baer [18]). Let N be a normal subgroup of G. Then N < ZU(G) if and only if N satisfies the Sylow tower property and Nq(P)/Cg(P) is strictly p-closed for all P G Sy\p(N) and p G w(N).
M. R. R. Moghaddam and M. A. Rostamvari (see [9]) introduced the concept of autonilpotent group. Let Ln(G) = {x G G | [x,a\,... ,an] = 1 Va\,...,an G AutG^. Then G is called autonilpotent if G = Ln(G) for some natural n. Some properties of autonilpotent groups were studied in [9]. In [8] all abelian autonilpotent groups were described. In particular, abelian autonilpotent non-unit groups of odd order don't exist. It was shown that if a ^group G is autonilpotent, then AutG is a p-group (Theorem 2.2 [10]). In [11, p. 45] it was asked: "Does there exist any odd order autonilpotent group?"
Corollary 2. Let, p be a prime. A p-group G is autonilpotent if and only if AutG is a p-group.
An example of a p-group G of order p5 (p > 3) such that AutG is also a p-group was constructed in [19]. In the library of small groups of GAP [20] there are 30 groups of order 36 such that their
3
the answer on the question from [11] is positive. From Theorem 2.3 [9] and Lemma 2.9 [10] it follows that a group is autonilpotent iff it is the direct product of its autonilpotent Sylow subgroups.
Corollary 3. A group G is autonilpotent if and only if it is the direct product of its Sylow subgroups and the automorphism group of a Sylow p-subgroup of G is a p-group for all p e n(G).
The proof of Corollary 3 doesn't use results from [9, 10]. The following result gives the description of elements in Z^(G, A).
Corollary 4. Let g be a p-element of a group G and InnG < A < AutG. Then g e Z^(G, A) if and only if ga = g for every p'-element a of A.
Corollary 5 (R. Baer [21]). Let p be a prime and G be a group. Then a p-element g of G belongs to Z^(G) if and only if it permutes with all p'-elements of G.
Corollary 6. A group G is autonilpotent if and only if every automorphism a of G fixes all elements of G whose orders are coprime to the order of a.
According to Frobenius p-nilpotencv criterion (see Theorem 5.26 [22, p. 171]) a group G is nilpotent if and only if NG(P)/CG(P) is a p-group for every p-subgroup P of G and every p e k(G).
Corollary 7. A group G is autonilpotent if and only if NAutG(P )/Cau\,g(P ) is a p-group for every p-subgroup P of G and every p e ■k(G).
2. Proves of the results
Lemma 1 (Lemma 3.6 [16]). Let P be a p-subgroup of a group G and R be a normal r-subgroup of G where r = p are primes. Then NG(P)R/R = Ng/r(PR/R), Gg{P)R/R = CG/R(PR/R) and NG(P)/CG(P) - Ng/r(PR/R)/Cg/r(PR/R).
Proof. [Proof of Theorem 1] Let prove Theorem 1 for A = InnG. In this case Z$(G, A) = (G).
Sufficiency. Let N be a normal subgroup of Z$(G) with the Svlow tower propertv. So N has a normal Svlow ^-subgroup Q. Note that Q < G and Q < Z$(G).
Hence G/Cg(Q) = NG(Q)/CG(Q) e F(q) by Lemma 2.5 from [23]. If Q = N, then sufficiency is proven.
Let Q<N. from N < ZF(G) it follows that N/Q < ZF(G/Q). Using induction on the order of G, we may assume that Ng/q(PQ/Q)/Cg/q(PQ/Q) e F(p) for every P e SylpN and p e k(N) \ Hence Ng(P)/Cg(P) - Ng/q(PQ/Q)/Cg/q(PQ/Q) e F(p) by Lemma 1. Thus sufficiency is proved.
Necessity. Let a group G be a minimal order counterexample with a normal subgroup N < Z$(G) that satisfies the statement of Theorem 1 and p be the greatest prime divisor of INThen a Svlow p-subgroup P of N is normal in ^^et H/K be a chief factor of G and H < P. Since CG(P) < CG(H/K) md G/Cg(P) = NG(P)/CG(P) e F(p), G/Cg(H/K) e F(p). It means that p < Zf(G). Note that NG/P(RP/P)/Cg/p(RP/P) - NG(R)/CG(R) e F(p) for every R e SylrN and r e n(N) \ {p} by Lemma 1. From IG/P| < |G| it follows that N/P < ZF(G/P). Thus N < Zf(G), the contradiction.
Assume now that InnG < A < Aut^^et r = G x ^^om InnG < A it follows that groups of automorphisms that are induced by A and r on a given section of G are isomorphic. It means that Na(H/K)/Ca(H/K) - Nr(H/K)/Cr(H/K) for a given section H/K oi G. In particular, every ^^^^^^sition A- F-central factor of G is an ^^^^^^^ chief fact or of T.
It means that ^^^^^^^^^ subgroup N of G that satisfies the Svlow tower property lies in ZF(G, A) if and onlv if N < ZF(r). The later is equivalent to Nr(P)/Cr(P) - NA(P)/CA(P) e F(p) for every P e SylpN and p e n(N). □
PROOF. [Proof of Corollary 1] Recall that U has the canonical local definition F where F(p) is the
class of all strictly p-closed groups, every sub group of Z&(G) is supersoluble and every supersoluble
□
Proposition 3. A group G is autonilpotent if and only if G = Z^(G, AutG).
Proof. [Proof\ Recall that the canonical local definition of N is F(p) = Np where Np is the class of all p-groups. Note that AutG/GAuto(Li(G)/Li-l(G)) ~ 1 £ F(p) to all prime p. Hence every AutG-composition factor of G between Li-l(G^d Li(G) is AutG-N-central. It means that Li(G) < Z^(G, AutG) for every i. So if G is autonilpotent group, then G = Z^(G, AutG). Assume that G = Z^(G, AutG). Hence there exists an AutG-composition series
1 = Go < • • • < Gn = G
with AutG-N-central chief factors. Let r = G x AutG. Note that
AutG/GAutc(Gi/Gi-i) ~ r/Cr(Gi/Gi-1) £ N
for every p £ ^(Gi/Gi-l) by analogy with the proof of Theorem 1. Note that r/Gr(Gi/Gi-l) does not have non-trivial ^subgroups for all p £ ^(Gi/Gi-l) by Lemma 3.9 [24, p. 26]. So AutG = GAutc(Gi/Gi-l^. Hence [Gi, AutG] < Gi-1. ft means that [x,al,... ,an] = 1 for all x £ G and al,... ,an £ AutG. Thus G = Ln(G) and G is autonilpotent. □
PROOF. [Proof of Corollary 3] Note that every nilpotent group satisfies the Svlow tower property and every autonilpotent group is nilpotent.
So a nilpotent group is autonilpotent if and only if NAutc(P)/CAutc(P) £ Np for every P £ Sy\pG and p £ ft(G) by Proposition 3 and Theorem 1. The automorphism group of a direct product of groups was described in [25]. In particular, if G = P x H, where P is a Svlow subgroup
of G, then AutG = AutP x AutH and NAutc(P)/CAutc(P) ^ AutP. □
□
Proof. [Proof of Corollary 4] From InnG < A ft follows that Z^(G,A) < Z^(G) is nilpotent. Hence every Svlow subgroup of Z^(G, A) is A-admissible. So
Na(P)/Ga(P) = A/Ca(P) £ N
for every P £ SylpZ^(G, l^d p £ (G, A)) by Theorem 1. It means that if g is a p-element of Z^(G, A), then ga = g for every p'-element a of A.
Let Gp be the set of all elements of G such that ga = g for every p'-element a of A and every g £ Gp. Note that if x,y £ Gp, then xy £ Gp. Hence Gp is a subgroup of G. Let g £ Gp, a, ft £ A
and ft be a p'-element. Then fta 1 is a p'-element too. Hence (g™)13 = gal3a la = g^a a = ga. It means that ga £ Gp. Thus Gp is an A-admissible subgroup of G. Let x be a p'-element of Gp. From InnG < A ft follows that ax : g ^ gx is a p'-element of A Hence ft acts trivially on Gp. It means that x < Z(GP). Let P £ Sy\pGp. Then P < Gp. So all p-elements of Gp form a subgroup P. Note that P is A-admissible and NA(P)/CA(P) £ Np. Therefore P < Z^(G, A) by Theorem 1. □ PROOF. [Proof of Corollary 5] Let g be a ^element of G. Note that xg = gx is equivalent to gx = g
and {ax : g ^ gx | x is a p'-element of G} is the set of all p'-elements of InnG. Now Corollary 5
□
□
PROOF. [Proof of Corollary 7] Assume that G is autonilpotent. Then
NAutc(P)/CAutG(P) £ Np
for every p-subgroup P of G and p £ n(G) by Corollary 6.
Assume now that NAutc(P)/CAutc(P) £ Np for everv p-subgroup P of G andp £ k(G). Suppose that G is non-nilpotent. So there is a Schmidt subgroup S of G. Then S has a normal ^-subgroup Q for some prime q and there is a ^'-element x of S with x £ Cs(Q) (see Theorem 26.1 [24, p. 243]). Since ax : g ^ gx is a non-identity inner automorphism of G of q'-order, NAutG(Q)/CAutG(Q) £ Nq, a contradiction.
Thus G is nilpotent. Hence G is autonilpotent by Proposition 3 and Theorem 1. □
Acknowledgments
I am grateful to A. F. Vasil'ev for helpful discussions.
REFERENCES
1. Kaloujnine L. 1953, "Uber gewisse Beziehungen zwischen einer Gruppe und ihren Automorphismen", Berliner Mathematische Tagung, Berlin, pp. 164-172.
2. Hall P. 1958, "Some sufficient conditions for a group to be nilpotent", Illinois J. Math., vol. 2, no. 4, pp. 787-801.
3. Huppert B. 1954, "Normalteiler und maximale Untergruppen endlicher Gruppen", Math. Z., vol. 60, no. 4, pp. 409-434.
4. Semetkov L.A. 1974, "Graduated formations of groups", Mathematics of the USSR-Sbornik, vol. 23, no. 4, pp. 593-611.
5. Schmid P. 1974, "Lokale Formationen endlicher Gruppen", Math. Z., vol. 137, no. 1, pp. 31-48.
6. Doerk K., Hawkes T. 1992, Finite soluble groups, Walter de Gruvter, Berlin-New York, 891 p.
7. Hegartv P. V. 1994, "The absolute center of a group", J. Algebra, vol. 169, pp. 929-935.
8. Nasrabadi M.M., Gholamiam A. 2014, "On A-nilpotent abelian groups", Proc. Indian Acad. Sei. (Math. Sei.), vol. 124, no. 4, pp. 517-525.
9. Davoudirad S., Moghaddam M.R. R., Rostamvari M. A. 2016, "Autonilpotent groups and their properties", Asian-European J. Math., vol. 9, no. 2, 1650056 (7 pages).
10. Hoseini S., Moghaddam M. R. R., Tajnia S. 2015, "On Auto-nipoltent groups", Southeast Asian Bull. Math., vol. 39, pp. 219-224.
11. Arora H., Karan R. 2018, "On Autonilpotent and Autosoluble Groups", Note Mat., vol. 38, no. 1, pp. 35-45.
12. Vasil'ev A. F., Vasil'eva T. I., Tvutvanov V. N. 2010, "On the finite groups of supersoluble type", Sib. Math. J., vol. 51, no. 6, pp. 1004-1012.
13. Monakhov V. S., Kniahina V.N. 2013, "Finite groups with P-subnormal subgroups", Ricerche mat., vol. 62, pp. 307-322.
14. Zimmermann I. 1989, "Submodular subgroups in finite groups", Math. Z., vol. 202, pp. 545-557.
15. Vasilvev V. A. 2015, "Finite groups with submodular Sylow subgroups", Siberian Math. J., vol. 56, no. 6, pp. 1019-1027.
16. Murashka V.l. 2018, "On analogues of Baer's theorems for widely supersoluble hvpercenter of finite groups", Asian-European J. Math., vol. 11, no. 3, 1850043 (8 pages).
17. Murashka V. I. 2014, "Properties of the class of finite groups with P-subnormal cyclic primary subgroups", Dokl. NAN Belarusi, vol. 58, no. 1, pp. 5-8. (In Russian)
18. Baer R. 1959, "Supersoluble immersion", Canad. J. Math., vol. 11, pp. 353-369.
19. Curran M.J. 1988, "Automorphisms of certain p-groups (p odd)", Bull. Austral. Math. Soc., vol. 38, pp. 299-305.
20. Groups, Algorithms, and Programming (GAP), Version 4.9.1 (2018). Available at http://www.gap-svstem.org (accessed 10 June 2018).
21. Baer R. 1953, "Group Elements of Prime Power Index", Trans. Amer. Math Soc., vol. 75, no. 1, pp. 20-47.
22. Isaacs I. M. 2008, Finite group theory, Graduate studies in mathematics, vol. 92, Providence, American Mathematical Society, 350 p.
23. Skiba A.N. 2012, "On the F-hvpercenter and the intersection of all F-maximal subgroups of a finite group", J. Pure Appl. Algebra, vol. 216, no. 4, pp. 789-799.
24. Shemetkov L. A. 1978, Formations of finite groups, Nauka, Moscow, 272 p. (In Russian)
25. Bidwell J.N.S., Curran M.J., McCaughan D.J. 2006, "Automorphisms of direct products of finite groups", Arch. Math., vol. 86, pp. 481-489.
СПИСОК ЦИТИРОВАННОЙ ЛИТЕРАТУРЫ
1. Kaloujnine L. Uber gewisse Beziehungen zwischen einer Gruppe und ihren Auto- morphismen // Berliner Mathematische Tagung, Berlin, 1953, P. 164-172.
2. Hall P. Some sufficient conditions for a group to be nilpotent // Illinois J. Math. 1958. V. 2, № 4. P. 787-801.
3. Huppert В. Normalteiler und maximale Untergruppen endlicher Gruppen // Math. Z. 1954. V. 60, № 4. P. 409-434.
4. Шеметков Л. А. Ступенчатые формации групп // Матем. сб. 1974. Т. 94(136), № 4(8), С. 628-648.
5. Schmid Р. Lokale Formationen endlicher Gruppen // Math. Z. 1974. V. 137, № 1. P. 31-48.
6. Doerk K., Hawkes T. Finite soluble groups. Berlin-New York: Walter de Gruvter, 1992. 891p.
7. Hegartv P. V. The absolute center of a group //J. Algebra. 1994. V. 169. P. 929-935.
8. Nasrabadi M.M., Gholamiam A. On A-nilpotent abelian groups // Proc. Indian Acad. Sei. (Math. Sei.). 2014. V. 124, № 4. P. 517-525.
9. Davoudirad S., Moghaddam M.R. R., Rostamvari M.A. Autonilpotent groups and their properties // Asian-European J. Math. 2016. V. 9, № 2. 1650056 (7 pages).
10. Hoseini S., Moghaddam M. R. R., Tajnia S. On Auto-nipoltent groups // Southeast Asian Bull. Math. 2015. V. 39. P. 219-224.
11. Arora H., Karan R. On Autonilpotent and Autosoluble Groups // Note Mat. 2018. V. 38, № 1. P. 35-45.
12. Васильев А.Ф., Васильева Т. И., Тютянов В.Н. О конечных группах сверхразрешимого типа // Сиб. мат. журн. 2010. Т. 51, № 6. С. 1270-1281.
13. Monakhov V. S., Kniahina V.N. Finite groups with P-subnormal subgroups // Ricerche mat. 2013. V. 62, P. 307-322.
14. Zimmermann I. Submodular subgroups in finite groups // Math. Z. 1989. V. 202. P. 545-557.
15. Васильев В. А. Конечные группы с субмодулярными силовскими подгруппами // Сиб. мат. журн. 2015. Т. 56, № 6. С. 1277-1288.
16. Murashka V. I. On analogues of Baer's theorems for widely supersoluble hvpercenter of finite groups // Asian-European J. Math. 2018. V. 11, Ж 3. 1850043 (8 pages).
17. Мурашко, В. И. Свойстав класса конечных групп с Р-субнормальными циклическими при-марными подгруппами // Докл. НАН Беларуси. 2014. Т. 58, № 1. С. 5-8.
18. Baer R. Supersoluble immersion // Canad. J. Math. 1959. V. 11. P. 353-369.
19. Curran M.J. Automorphisms of certain p-groups (p odd) / / Bull. Austral. Math. Soc. 1988. V. 38. P. 299-305.
20. Groups, Algorithms, and Programming (GAP), Version 4.9.1. [Электронный ресурс] // URL: http://www.gap-svstem.org (дата обращения 10.06.2018).
21. Baer R. Group Elements of Prime Power Index // Trans. Amer. Math Soc. 1953. V. 75, № 1. P. 20-47.
22. Isaacs I. M. Finite group theory. Graduate studies in mathematics, V. 92. Providence: American Mathematical Society, 2008. 350 p.
23. Skiba A.N. On the F-hvpercenter and the intersection of all F-maximal subgroups of a finite group //J. Pure Appl. Algebra. 2012. V. 216, № 4. P. 789-799.
24. Шеметков Л. А. Формации конечных групп. М.: Наука, 1978. 272 с.
25. Bidwell J.N.S., Curran M.J., McCaughan D.J. Automorphisms of direct products of finite groups // Arch. Math. 2006. V. 86. P. 481-489.
Получено 15.06.2018 г.
Принято в печать 12.07.2019 г.
