Petro Mykhailovych Gudivok
We present a brief scientific biography of Professor Petro M. Gudivok.
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irk-123456789-1523412019-06-11T01:25:14Z Petro Mykhailovych Gudivok Babych, M. Bondarenko, V.M. Drozd, Yu. Kirichenko, V. Shapochka, I. We present a brief scientific biography of Professor Petro M. Gudivok. 2013 Article Petro Mykhailovych Gudivok / M. Babych, V.M. Bondarenko, Yu. Drozd, V. Kirichenko, I. Shapochka // Algebra and Discrete Mathematics. — 2013. — Vol. 16, № 2. — С. 151–159. — Бібліогр.: 95 назв. — англ. 1726-3255 http://dspace.nbuv.gov.ua/handle/123456789/152341 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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We present a brief scientific biography of Professor Petro M. Gudivok. |
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Babych, M. Bondarenko, V.M. Drozd, Yu. Kirichenko, V. Shapochka, I. Petro Mykhailovych Gudivok Algebra and Discrete Mathematics |
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Babych, M. Bondarenko, V.M. Drozd, Yu. Kirichenko, V. Shapochka, I. |
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Babych, M. |
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Petro Mykhailovych Gudivok |
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Petro Mykhailovych Gudivok |
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Petro Mykhailovych Gudivok |
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Petro Mykhailovych Gudivok |
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Petro Mykhailovych Gudivok |
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petro mykhailovych gudivok |
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Petro Mykhailovych Gudivok / M. Babych, V.M. Bondarenko, Yu. Drozd, V. Kirichenko, I. Shapochka // Algebra and Discrete Mathematics. — 2013. — Vol. 16, № 2. — С. 151–159. — Бібліогр.: 95 назв. — англ. |
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Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics SURVEY ARTICLE
Volume 16 (2013). Number 2. pp. 151 – 159
c© Journal “Algebra and Discrete Mathematics”
Petro Mykhailovych Gudivok
Mykhailo Babych, Vitaliy M. Bondarenko, Yuriy Drozd,
Vladimir Kirichenko, Ihor Shapochka
Abstract. We present a brief scientific biography
of Professor Petro M. Gudivok.
Petro M. Gudivok
(April 12, 1936 – June 11, 2012)
Petro M. Gudivok was born in the
small village of Lavky, near Mukachevo
(Czechoslovakia, now Ukraine) in a large
peasant family. His childhood coincided
with the difficult war period and the early
postwar years, making him accustomed
from an early age to the hard peasant
labor and responsibility for himself and
his family. He has graduated from the
Mukachevo Secondary School No. 2, hav-
ing studied there with great and talented
teachers. They have developed the boy’s
remarkable abilities, wit and inquiring
mind, as well as nurtured his love for
the Queen of Sciences, Mathematics, that
would become his life calling.
In 1953, having graduated from the
Secondary School with high honors, Petro M. Gudivok has started his
undergraduate studies at the Mathematics Department of Uzhhorod
State University, where he has achieved his first scientific results under the
guidance of Samuil D. Berman. In 1958, after graduating with honors from
Uzhhorod State University, he has worked for one year as an assistant at the
152 Petro Mykhailovych Gudivok
Chair of General Mathematics and in 1959 he started his graduate studies
under the supervision of the young and extremely talented algebraist
Samuil D. Berman. The range of research interests of Samuil D. Berman
during this period included one of the most promising parts of algebra —
group representation theory. Petro M. Gudivok has made a significant
contribution to this theory.
Starting in September of 1962, Petro M. Gudivok has occupied various
teaching positions at the Uzhhorod State University while at the same
time continuing with active research work. In 1963 he has defended his
Candidate Thesis "Representations of finite groups over some local rings"
at the Leningrad State Pedagogical Institute, followed by the defence of his
Doctoral Thesis "Representations of finite groups over local rings" at the
Leningrad State University in 1975. From 1980 until the last days of his life
Petro M. Gudivok was the Head of the Chair of Algebra of the Uzhhorod
National University. From 1986 till 1992 he occupied the position of the
Dean of the Department of Mathematics at the Uzhhorod State University.
Petro M. Gudivok died on June 11, 2012, after a long-lasting illness.
Petro M. Gudivok has developed the theory of representations of
finite groups over commutative local rings, created effective methods for
the study of the representations of finite groups over discrete valuation
rings, resolved a number of known problems on the number and degrees
of nonequivalent irreducible and indecomposable matrix representations
of finite groups over commutative local rings, on the wildness of the
problem of description of nonequivalent matrix representations of finite
groups over complete discrete valuation rings, on the reasonableness of the
Krull-Schmidt theorem for representations of finite groups over discrete
valuation rings.
Together with Vyacheslav P. Rud’ko he has obtained fundamental
results on tensor products of representations of finite groups over commu-
tative rings. They have also solved the problem on semisimplicity of the
algebra of representations of finite groups over P -adic integers over an
arbitrary field.
Petro M. Gudivok has successfully applied his methods from the
theory of representations of finite groups over commutative rings to the
study of linear groups over principal ideal domains, multi-dimensional
crystallographic groups and Chernikov groups. He has solved the problem
of conjugation of Sylov p-subgroups of the general linear group over the ring
of integers, showed that the problem of describing up to an isomorphism
the n-dimensional crystallographic groups for an arbitrary natural n is
wild. He has presented numerous reports at international congresses of
Babych, Bondarenko, Drozd, Kirichenko, Shapochka 153
mathematicians and at international and national conferences. He was
one of the leaders of research projects on the theory of representations of
groups and their applications that were carried out jointly between the V.
M. Glushkov Institute of Cybernetics and the Institute of Mathematics of
the National Academy of Sciences of Ukraine. Petro M. Gudivok published
more than 170 scientific papers and four monographs [1, 2, 3, 4] on the
theory of representations of finite groups over commutative rings and
their applications.
In addition to his prolific research work, the scientist was also actively
engaged in teaching and especially in finding and engaging talented
students in research. Professor Petro M. Gudivok has served as an advisor
on twelve Candidate of Science theses, while three of his students became
Doctors of Science.
Petro M. Gudivok has put a lot of effort and energy into his scientific
and organizational work: he was a member of the Scientific Council on
thesis defence at the Institute of Mathematics of the National Academy
of Sciences of Ukraine, as well as serving as an opponent and reviewer of
many Candidate and Doctoral theses in algebra, organizing a number of
conferences and seminars on algebra at the Uzhhorod National University
and serving as the chief editor of the "Scientific Bulletin of the Uzhhorod
University. Series Mathematics and Informatics" journal.
Scientific achievements of Professor Petro M. Gudivok have received a
high public and social recognition. He was awarded the title of Honorary
Worker of Education of Ukraine (1991), was elected Academician of the
Higher School of Ukraine (1992), received the title of Soros Professor
(1997). In 2005 for his many years of productive scientific and educational
work Petro M. Gudivok was awarded the title of Honored Professor of
the Uzhhorod National University.
The cherished memory of Petro M. Gudivok will forever remain in
the minds and hearts of his colleagues, students, family and friends. May
he rest in peace on his native Transcarpathian land.
List of Petro M. Gudivok’s key publications
[1] P. M. Gudivok, Integral reprsentations of finite groups, Uzhgorod. Univ., 1978.
[2] P. M. Gudivok, V. P. Rud’ko, Tensor products of representations of finite groups,
Uzhgorod. Univ., 1985.
[3] P. M. Gudivok, Reprsentations of finite groups over commutative local rings,
Uzhgorod. Nat. Univ., 2003.
[4] P. M. Gudivok, V. P. Rud’ko, V. A. Bovdi, Crystallographic groups, Uzhgorod.
Nat. Univ., 2006.
154 Petro Mykhailovych Gudivok
[5] P. M. Gudivok, On isomorphism of group algebras of cyclic extensions of super-
solable groups, Dokl. i Soobshch. Uzhgorod. Univ., Ser. Phis.-Mat. Nauk, 3, 1960,
pp. 58–59.
[6] P. M. Gudivok, On integral reprsentations of a finite group with noncyclic Sylow
p-subgroup, Uspekhi Mat. Nauk, 16, No. 2, 1961, pp. 229-230.
[7] S. D. Berman, P. M. Gudivok, Integral representations of finite groups, Dokl. Akad.
Nauk SSSR, 145, 1962, pp. 1199–1201.
[8] P. M. Gudivok, On p-adic integral representations of finite groups, Dokl. Uzhgorod
Univ. Ser. Phys.-Math. Nauk, 5, 1962, pp. 81–82.
[9] P. M. Gudivok, On relation of orders of conjugacy classes and degrees of absolutly
irreducible representations for certain classes of groups, Izv. vuzov "Matematika",
2, 1963, pp. 44–52.
[10] S. D. Berman, P. M. Gudivok, Indecomposable representations of finite groups over
the ring of p-adic integers, Izv. Akad. Nauk SSSR Ser. Mat., 28, 1964, pp. 875–910.
[11] P. M. Gudivok, V. S. Drobotenko, A.I. Lichtman, On representations of finite
groups over the ring of residue classes mod m, Ukrain. Mat. Zh., 16, 1964, pp. 82–89.
[12] P. M. Gudivok, On representations of finite groups over some local rings, Dopovidi
Akad. Nauk URSR, 2, 1964, pp. 173–176.
[13] P. M. Gudivok, Representations of finite groups over quadratic rings, Dokl. Akad.
Nauk SSSR, 6, 1964, pp. 1210–1213.
[14] P. M. Gudivok, Representations of finite groups over local number rings, Dopovidi
Akad. Nauk URSR, 8, 1966, pp. 979–981.
[15] P. M. Gudivok, V. P. Rud’ko On p-adic integral representations of the cyclic
p-group, Dopovidi Akad. Nauk URSR, 9, 1966, pp. 1111–1113.
[16] P. M. Gudivok, V. S. Drobotenko Extensions of divisible abelian groups by a cyclic
group, Dopovidi Akad. Nauk URSR, 10, 1966, pp. 1239–1242.
[17] P. M. Gudivok, Representations of finite groups over number rings, Izv. Akad.
Nauk SSSR, 4, 1967, pp. 799–834.
[18] L. F. Barannik, P. M. Gudivok,On projective representations of finite groups over
rings, Dopovidi Akad. Nauk URSR Ser. A, 4, 1968, pp. 294–297.
[19] L. F. Barannik, P. M. Gudivok, On indecomposable projective representations of
finite groups, Dopovidi Akad. Nauk URSR Ser. A, 5, 1969, pp. 391–393.
[20] P. M. Gudivok, E.Ya. Pogorilyak, The Krull-Schmidt theorem for group represen-
tations over P -integer number rings, Mat. Zametki, 7, 1970, pp. 125–135.
[21] L. F. Barannik, P. M. Gudivok, Projective representations of finite groups over
number rings, Mat. Sbornik, 82, 1970, pp. 423–443.
[22] P. M. Gudivok, The boundedness of the degrees of indecomposable modular rep-
resentations of finite groups over principal ideal domains, Dopovidi Akad. Nauk
URSR Ser. A, 8, 1971, pp. 683–685.
[23] P. M. Gudivok, S. F. Goncharova, V. P. Rud’koOn the algebra of integral p-adic
representations of a finite group, Dokl. Akad. Nauk SSSR, 198, 1971, pp. 509–512.
[24] P. M. Gudivok, On the Krull-Sahmidt theorem for 2-integer rational representations
of 2-group, Dopovidi Akad. Nauk URSR Ser. A, 2, 1972, pp. 112–114.
Babych, Bondarenko, Drozd, Kirichenko, Shapochka 155
[25] L. F. Barannik, P. M. Gudivok, On algebra of projective integral representations
of finite groups, Dopovidi Akad. Nauk URSR Ser. A, 4, 1972, pp. 291–293.
[26] P. M. Gudivok, On the number of indecomposable integral p-adic representations
of twisted group rings, Mat. Sbornik, 91, 1973, pp. 27–49.
[27] P. M. Gudivok, V. P. Rud’koOn modular and integral representation algebras of
finite groups, Izv. Akad. Nauk SSSR, 37, 1973, pp. 963–987.
[28] P. M. Gudivok, S. F. Goncharova, V. P. Rud’koFinite-dimensional algebras of
integral P -adic representations of a finite group, Mat. Sbornik, 94, 1974, pp. 358–
384.
[29] P. M. Gudivok, Some observations on integral representations of finite groups,
Ukrain. Mat. Zh., 26, 1974, pp. 539–545.
[30] P. M. Gudivok, On modular and integral representations of finite groups, Dokl.
Akad. Nauk SSSR, 214, 1974, pp. 993–996.
[31] L. F. Barannik, P. M. Gudivok, On the number of indecomposable integral reprsen-
tations of twisted group rings of finite groups and rings of p-adic integers, Math.
Collect., Acad. Sci. Ukr. SSR, Inst. Math., 1976, pp. 164–167.
[32] P. M. Gudivok, On modular and integral P-adic reprsentations of the direct prtoduct
of groups, Ukrain. Mat. Zh., 29, 1977, pp. 580–588.
[33] P. M. Gudivok, On reprsentations of a direct prtoduct of groups over complete
discrete valuation rings, Dokl. AN SSSR, 237, 1977, pp. 25–27.
[34] P. M. Gudivok, On reprsentations of finite groups over complete discrete valuation
rings, Trudy Mat. Inst. Steklova, 148, 1978, pp. 96–105.
[35] L. F. Barannik, P. M. Gudivok, Crossed group rings of finite groups and rings of
p-adic integers with finitely many indecomposable integral representations, Mat.
Sbornik, 108, 1979, pp. 187–211.
[36] P. M. Gudivok, A. A. Kirilyuk, The Sylow p-subgroups of the general linear group
over discrete valuation rings, Dopovidi Akad. Nauk URSR Ser. A, 5, 1979, pp.
326–328.
[37] P. M. Gudivok, E. Ya. Pogorilyak, On reprsentations of finite groups over some
local rings, Dopovidi Akad. Nauk URSR Ser. A, 5, 1981, pp. 326–328.
[38] P. M. Gudivok, S. F. Goncharova, V. P. Rud’ko, On tensor products of integral
p-adic representations of finite groups, Ukrain. Mat. Zh., 4, 1982, pp. 688–694.
[39] P. M. Gudivok, A. A. Kirilyuk, V. P. Rud’ko, A. I. Tsitkin, On finite subgroups of
GL(n,Z), Kibernetika, 6, 1982, pp. 71–82.
[40] A. A. Bovdi, P. M. Gudivok, M. S. Semirot, Normal group rings, Ukrain. Mat.
Zh., 4, 1982, pp. 688–694.
[41] V. F. Barannik, P. M. Gudivok, V. P. Rud’ko, On tensor products of representations
of finite groups over complete discrete valuation rings, Dopovidi Akad. Nauk URSR
Ser. A, 4, 1985, pp. 9–12.
[42] P. M. Gudivok, Yu. V. Kapitonova, On finite irreducible subgroups of GL(n,Z),
Kibernetika, 5, 1986, pp. 1–16.
[43] P. M. Gudivok, F. G. Vashchuk, Integral p-adic representations of finite abelian
p-groups, Dopovidi Akad. Nauk URSR Ser. A, 1, 1986, pp. 3-6.
156 Petro Mykhailovych Gudivok
[44] P. M. Gudivok, T. M. Sapuzhak, Irreducible representations of finite groups over
principal ideal domains, Dopovidi Akad. Nauk URSR Ser. A, 6, 1987, pp. 3–6.
[45] P. M. Gudivok, V.P. Rud’ko, On tensor products of irreducible representations of
finite groups over complete discrete valuation rings, Rings and moduls. Limited
theorems of probability theory, Leningrad. Univ., 1988, pp. 50–68.
[46] P. M. Gudivok, E.Ya. Pogorilyak, Representations of finite p-groups over local
rings of positive characteristic, Dopovidi Akad. Nauk URSR Ser. A, 2, 1989, pp.
5–8.
[47] P. M. Gudivok, Yu. V. Kapitonova, S. S. Polyak, V. P. Rud’ko, A.I. Tsitkin,
Conjugacy classes of a unitriangle group, Kibernetika, 1, 1990, pp. 40–48.
[48] P. M. Gudivok, E.Ya. Pogorilyak, On modular representations of finite groups
over integral domains, Trudy Mat. Ins. Akad Nauk SSSR, 183, 1990, pp. 78–86.
[49] P. M. Gudivok, V. M. Oros, A. V. Roiter, On representations of finite groups over
a ring of formal power series with p-adic integer coefficients, Dokl. Akad. SSSR,
314, 1990, pp. 49–52.
[50] P. M. Gudivok, On Sylow p-subgroups of the general linear group over the ring of
integers, Algebra i Analiz, 2, 1990, pp. 125–131.
[51] P. M. Gudivok, On Sylow p-subgroups of the general linear group over the complete
discrete valuation rings, Ukrain. Mat. Zh., 43, 1991, pp. 918–924.
[52] A. F. Barannik, P. M. Gudivok, The ring of projective integral p-adic represen-
tations of a finite group, Mat. Zbirnik Mat. Prats Lviv. Mat. Tov., 1, 1991, pp.
44–54.
[53] P. M. Gudivok, On representations of finite groups over some factorial rings, Cont.
Math., 131, 1992, pp. 173–181.
[54] P. M. Gudivok, V. M. Oros, A. V. Roiter, On representations of finite p-groups
over a ring of formal power series with p-adic integer coefficients, Ukrain. Mat.
Zh., 44, 1992, pp. 753–765.
[55] P. M. Gudivok, F. G.Vashchuk, V. S.Drobotenko, Chernikov p-groups and integral
p-adic representations of finite groups, Ukrain. Mat. Zh., 44, 1992, pp. 742–753.
[56] P. M. Gudivok, V. P. Rud’ko, On isomorphism of Sylow p-subgroups of general
linear group over the ring of integers, Dopovidi Akad. Nauk Ukraini, 9, 1992, pp.
3–5.
[57] P. M. Gudivok, V. P. Rud’ko, On tensor products of representations of finite groups
over some complete local rings, Dopovidi Akad. Nauk Ukraini, 8, 1993, pp. 5–7.
[58] P. M. Gudivok, On the equivalence of matrix over commutative rings, Infinite
groups and related algebraic structures, Institut Matematiki AN Ukrainy, 1993,
pp. 431–437.
[59] V. M. Bondarenko, P. M. Gudivok, On representations of finite p-groups over a
ring of formal power series with p-adic integer coefficients, Infinite groups and
related algebraic structures, Institut Matematiki AN Ukrainy, 1993, pp. 5–14.
[60] V. F. Barannik, P. M. Gudivok, V. P. Rud’ko, On tensor products of irreducible
integral P -adic representations of finite p-groups, Nauk. Visnik Uzhgorod. Univ.
Ser. Mat., 1, 1994, pp. 5–13.
Babych, Bondarenko, Drozd, Kirichenko, Shapochka 157
[61] P. M. Gudivok, I. V. Shapochka, On extensions of abelian groups, Dopovidi Nats.
Akad. Nauk Ukraini, 2, 1995, pp. 8–9.
[62] P. M. Gudivok, V. P. Rud’ko, On Sylow subgroups of the general linear group over
integral domains, Dopovidi Nats. Akad. Nauk Ukraini, 8, 1995, pp. 5–7.
[63] P. M. Gudivok, V. I. Pogorilyak, On indecomposable representations of finite
p-groups over commutative local rings, Dopov. Akad. Nauk Ukr., 5, 1996, pp. 7–11.
[64] P. M. Gudivok, V. S. Drobotenko, Samuil Davidovich Berman, Nauk. Visn. Uzh-
gorod. Univ., Ser. Mat. 2, 1997, pp. 4–11.
[65] P. M. Gudivok, V. P. Rud’ko, Irreducible Sylow p-subgroups of general linear
groups over the ring of integers, Nauk. Visn. Uzhgorod. Univ., Ser. Mat. 2, 1997,
pp. 29–37.
[66] P. M. Gudivok, T. M. Sapuzhak, , Irreducible representations of finite groups over
p-adic quadratic rings, Nauk. Visn. Uzhgorod. Univ., Ser. Mat. 2, 1997, pp. 38–44.
[67] P. M. Gudivok, I. V. Shapochka, On the wildness of the description problem for
some classes of groups, Nauk. Visn. Uzhgorod. Univ., Ser. Mat. 3, 1998, pp. 69–77.
[68] P. M. Gudivok, O. A. Tylyshchak, On the irreducible modular representations of
finite p-groups over commutative local rings, Nauk. Visn. Uzhgorod. Univ., Ser.
Mat. 3, 1998, pp. 78-83.
[69] P. M. Gudivok, M. G. Ivanov, On the irreducible matrix representations of finite
p-groups over local factorial rings, Nauk. Visn. Uzhgorod. Univ., Ser. Mat. 3, 1998,
pp. 84-88.
[70] P. M. Gudivok, On representations of twisted group rings of finite groups over
rings of p-adic integers, Dopov. Nats. Akad. Nauk Ukr., Mat. Pryr. Tekh. Nauky,
7, 1998, pp. 19–23.
[71] P. M. Gudivok, V. I. Pogorilyak, On indecomposable matrix representations of finite
p-groups over commutative local rings of characteristic ps, Nauk. Visn. Uzhgorod.
Univ., Ser. Mat., 4, 1999, pp. 43–46.
[72] P. M. Gudivok, I. P. Sygetij, I. B. Chukhraj, On the number of matrix repre-
sentations of fixed degree of a finite p-group over some commutative rings of
characteristic ps, Nauk. Visn. Uzhgorod. Univ., Ser. Mat., 4, 1999, pp. 47–53.
[73] P. M. Gudivok, I. V. Shapochka, On the Chernikov p-groups, Ukrain. Mat. Zh.,
51, No. 3, 1999, pp. 291–304.
[74] P. M. Gudivok, I. B. Chukhraj, On the number of indecomposable matrix rep-
resentations of given degree of a finite p-group over commutative local rings of
characteristic ps, Nauk. Visn. Uzhgorod. Univ., Ser. Mat. i Inf., 5, 2000, pp. 33–40.
[75] P. M. Gudivok, I. V. Shapochka, On the irreducible matrix representations of
finite 2-groups over local factorial rings, Nauk. Visn. Uzhgorod. Univ., Ser. Mat. i
Inf., 5, 2000, pp. 103–105.
[76] P. M. Gudivok, I. B. Chukhraj, On the number of nonequivalent indecomposable
matrix representations of the given degree of a finite p-group over commutative
local ring of characteristic ps, An. Ştiinţ. Univ. “Ovidius" Constanţa, Ser. Mat., 8,
no. 2, 2000, pp. 27–36.
[77] P. M. Gudivok, V. P. Rud’ko, On isomorphism of Sylow subgroups of the general
linear group over the ring of integers, J. Math. Sci., New York, 102, No. 3, 2000,
pp. 3998-4008.
158 Petro Mykhailovych Gudivok
[78] P. M. Gudivok, V. P. Rud’ko, N. V. Yurchenko, On Sylow p-subgroups of the
general linear group over a principal ideal domain, Nauk. Visn. Uzhgorod. Univ.,
Ser. Mat. i Inf., 6, 2001, pp. 31–46.
[79] V. A. Bovdi, P. M. Gudivok, V. P. Rud’ko, Torsion-free groups with indecomposable
holonomy group. I, J. Group Theory, 5, No. 1, 2002, pp. 75–96.
[80] P. M. Gudivok, V. I. Pogorilyak, Matrix representations of finite p-groups over
commutative local rings of characteristic ps, Ukrain. Mat. Zh. 54, No. 6, 2002,
pp. 764–770.
[81] V. A. Bovdi, P. M. Gudivok, V. P. Rud’ko, On torsion-free crystallographic groups
with indecomposable point groups, Ukrainian mathematical congress – 2001, Kyiv,
Ukraine, August 21–23, 2001. Proceedings. Section 1. Algebra and number theory,
Instytut Matemat. NAN Ukrainy, 2002, pp. 5–18.
[82] P. M. Gudivok, On development of the theory of representations of finite groups
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Contact information
M. Babych Chair of Intellectual Information Technologies,
Department of International Economics and
Management, Ukrainian State University of Fi-
nance and International Trade, Chygorin Street
57, Kyiv, Ukraine 01601
E-Mail: myhailo.babych@gmail.com
V. M. Bondarenko Institute of Mathematics, National Academy of
Sciences of Ukraine, Tereschenkivska 3, 01601
Kyiv, Ukraine
E-Mail: vit-bond@imath.kiev.ua
URL: http://www.imath.kiev.ua
Yu. Drozd Institute of Mathematics, National Academy of
Sciences of Ukraine, Tereschenkivska 3, 01601
Kyiv, Ukraine
E-Mail: y.a.drozd@gmail.com,
drozd@imath.kiev.ua
URL: www.imath.kiev.ua/∼drozd
V. V. Kirichenko Department of Geometry, School of Mechan-
ics and Mathematics, Kyiv National Taras
Shevchenko University, Volodymyrska, 64, Kyiv,
01033, Ukraine
E-Mail: vkir@univ.kiev.ua
I. Shapochka Chair of Algebra, Department of Mathematics,
Uzhhorod National University, Universitetska
str, 14, 88000, Uzhhorod, Ukraine
E-Mail: kaf-algebra@uzhnu.edu.ua
URL: uzhnu.edu.ua
Received by the editors: 17.10.2013
and in final form 03.11.2013.
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