On minimal non- MSP -groups
A finite group $G$ is called an $MSP$-group if all maximal subgroups of the Sylow subgroups of $G$ are Squasinormal in $G$. In this paper, wc give a complete classification of those groups which are not $MSP$-groups but whose proper subgroups are all $MSP$-groups.
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2011
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508780008046592 |
|---|---|
| author | Guo, P. Zhang, Xirong Го, П. Чжан, Хіронг |
| author_facet | Guo, P. Zhang, Xirong Го, П. Чжан, Хіронг |
| author_sort | Guo, P. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:36:55Z |
| description | A finite group $G$ is called an $MSP$-group if all maximal subgroups of the Sylow subgroups of $G$ are Squasinormal in $G$. In this paper, wc give a complete classification of those groups which are not $MSP$-groups but whose proper subgroups are all $MSP$-groups. |
| first_indexed | 2026-03-24T02:30:38Z |
| format | Article |
| fulltext |
К О Р О Т К I П О В I Д О М Л Е Н Н Я
UDC 512.5
P. Guo (Lianyungang Teachers College; School Appl. Sci., Taiyuan Univ. Sci. and Technol., China),
X. Zhang (Shanghai Univ., China)
ON MINIMAL NON-MSP -GROUPS*
ПРО МIНIМАЛЬНI НЕ MSP -ГРУПИ
A finite group G is called an MSP -group if all maximal subgroups of the Sylow subgroups of G are S-
quasinormal in G. In this paper, wc give a complete classification of those groups which are not MSP -groups
but whose proper subgroups are all MSP -groups.
Скiнченну групу G називають MSP -групою, якщо всi максимальнi пiдгрупи силовських пiдгруп G є
S-квазiнормальними в G. Наведено повну класифiкацiю груп, якi не є MSP -групами, але всi їх власнi
пiдгрупи є MSP -групами.
1. Introduction. In this paper, only finite groups are considered and our notation is
standard.
Two subgroups H and K of a group G are said to permute if HK = KH. A
subgroup H of G is called quasinormal in G if it permutes with every subgroup of G.
Kegel [9] called a subgroup H of G S-quasinormal in G if it permutes with every Sylow
subgroup of G. Srinivasan [14] studied groups in which all maximal subgroups of the
Sylow subgroups are S-quasinormal subgroups and we call such groups MSP -groups.
The study of the structure of groups which have some kind of property has attracted
much attention in group theory and many meaningful results about this topic have been
obtained. For example, Schmidt [13] determined the structure of minimal non-nilpotent
groups, and Doerk [6] [determined the structure of minimal non-supersolvable groups.
These achievements have indeed pushed forward the developments of group theory. The
further results can consult [2, 12, 15].
In this paper, we call a groupG a minimal non-MSP -group if every proper subgroup
of G is an MSP -group but G itself is not and the minimal non-MSP -groups are
classified completely.
2. Preliminary results. We collect some lemmas which will be frequently used in
the sequel.
Lemma 1 ([14], Theorem 2). If a group G is an MSP -group, then G is supersolv-
able.
Lemma 2. Let G be a minimal non-MSP -group. Then there exists a normal
Sylow p-subgroup P of G and a non-normal Sylow q-subgroup Q of G with p 6= q such
that |G| = paqb and at least one of a and b is more than 1.
Proof. Since every proper subgroup of G is an MSP -group, G is supersolvable or
minimal non-supersolvable by Lemma 1. So G is solvable.
Let {P1, P2, . . . , Ps} is a Sylow basis of G. Since G is a minimal non-MSP -group,
there exists i and a maximal subgroup P ∗ of Pi such that P ∗ is not S-quasinormal in G.
*The research of the work was partially supported by the National Natural Science Foundation of China
(No. 11071155), SGRC(GZ310) and Special Foundation from Department of Land and Resources of Shanxi
Province of China (No. 0905910).
c© P. GUO, X. ZHANG, 2011
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 9 1279
1280 P. GUO, X. ZHANG
If s ≥ 3, then PiPj (j = 1, 2, . . . , i− 1, i+ 1, . . . , s) is a proper subgroup of G. Since
every proper subgroup of G is an MSP -group, it follows that P ∗ is S-quasinormal in
PiPj . Applying [3] (Lemma 2.6), P ∗ is normal in PiPj . Therefore, P ∗ is normal in G,
a contradiction. Hence |G| = paqb and at least one of a and b is more than 1.
The following fact follows from [31] (Lemma 2.6) and [16] (Theorem 6).
Lemma 3. A group G is an MSP -group if and only if G = H o 〈x〉 where:
(i) H is a normal nilpotent Hall subgroup of G;
(ii) every generator of every Sylow subgroup of 〈x〉 induces a power automorphism
of order dividing a prime in H/Φ(H).
Lemma 4 ([10], 13.4.3). Let α be a power automorphism of an abelian group A.
If A is a p-group of finite exponent, then there is a positive integer l such that aα = al
for all a in A. lf α is nontrivial and has order prime to p, then α is fixed-point-free.
Lemma 5 [8]. Let {P1, P2, . . . , Pr} is a Sylow basis of a solvable group G. Then
the following statements are equivalent:
(a) Every subgroup of Pi permutes with every subgroup of Pj for i 6= j.
(b) The nilpotent residual N of G is an abelian Hall subgroup of G and every
element of G induces a power automorphism in N.
Lemma 6 ([7], Lemma 2.9). If a p-group G of order pn+1 has a unique non-cyclic
maximal subgroup, then G is isomorphic to one of the following groups:
(I) Cpn × Cp = 〈a, b | apn = bp = 1, [a, b] = 1〉, where n ≥ 2;
(II) Mpn+1 = 〈a, b | apn = bp = 1, b−1ab = a1+p
n−1〉, where n ≥ 2 and n ≥ 3 if
p = 2.
Lemma 7 [6]. Let G be a minimal non-supersolvable group. Then:
(1) G has only one normal Sylow p-subgroup P ;
(2) P/Φ(P ) is a minimal normal subgroup of G/Φ(P ) and P/Φ(P ) is non-cyclic;
(3) if p 6= 2, then the exponent of P is p;
(4) if P is non-abelian and p = 2, then the exponent of P is 4;
(5) if P is abelian, then the exponent of P is p.
3. Main results. In this section, we give the complete classification of minimal
non-MSP -groups.
Theorem 1. Let p and q are distinct prime divisors of the order of a group G.
Then G is a minimal non-MSP -group if and only if G is of one of the following types:
(I) G = 〈x, y | xp = yq
n
= 1, y−1xy = xr〉, where rq 6≡ 1 (mod p), rq
2 ≡
≡ 1 (mod p), q | p− 1, n ≥ 2 and 0 < r < p;
(II) G = P oQ, where P = 〈a, b〉 is an elementary abelian p-group of order p2 and
Q = 〈y〉 is cyclic of order qr; define ay = ai, by = bi
j
, p ≡ 1 (mod q) and r ≥ 1, where
i is the least positive primitive q-th root of unity modulo p, j = 1 + k and 0 < k < q;
(III) G = 〈x, y | x4p = 1, y2 = x2p, y−1xy = x−1〉;
(IV) G = 〈x, y, z | xp = yq
n−1
= zq = 1, y−1xy = xr, [x, z] = 1, [y, z] = 1〉,
where r 6≡ 1 (mod p), rq ≡ 1 (mod p) and n ≥ 3;
(V) G = 〈x, y, z | xp = yq
n−1
= zq = 1, y−1xy = xr, [x, z] = 1, z−1yz =
= y1+q
n−2〉, where r 6≡ 1 (mod p), rq ≡ 1 (mod p), n ≥ 3 and n ≥ 4 if q = 2;
(VI) G = P oQ, where Q = 〈y〉 is cyclic of order qr > 1, with q - p− 1, and P is
an irreducible Q-module over the field of p elements with kernel 〈yq〉 in Q;
(VII) G = P oQ, where P is a non-abelian special p-group of rank 2m, the order
of p modulo q being 2m, Q = 〈y〉 is cyclic of order qr > 1, y induces an automorphism
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ON MINIMAL NON-MSP -GROUPS 1281
in P such that P/Φ(P ) is a faithful and irreducible Q-module, and y centralizes Φ(P );
furthermore, |P/Φ(P )| = p2m and |P ′| ≤ pm;
(VIII) G = P oQ, where P = 〈a0, a1, . . . , aq−1〉 is an elementary abelian p-group
of order pq, Q = 〈y〉 is cyclic of order qr, q is the highest power of q dividing p − 1
and r > 1. Define ayj = aj+i for 0 ≤ j < q − 1 and ayq−1 = ai0, where i is a primitive
q-th root of unity modulo p.
Proof. If G is a minimal non-MSP -group, then we may assume G = PQ with
P E G and Q 5 G by Lemma 2, where P ∈ Sylp(G) and Q ∈ Sylq(G). Since all the
Sylow q-subgroups are conjugate in G, we only consider the case that Q acts on P.
Assume that neither P nor Q is cyclic. For two maximal subgroups Q1 and Q2 of
Q and every maximal subgroup P1 of P, since PQ1 and PQ2 are MSP -groups, it
follows from Lemma 3 that P1 is normal in not only PQ1 but also PQ2. Hence P1 is
normal in G. For two maximal subgroups A and B of P, we have that AQ and BQ
are MSP -groups. By Lemma 3, Q is normal in G = 〈A,B,Q〉, a contradiction. Hence
either P or Q is cyclic.
Now we divide into the following four cases to discuss.
(1) Assume that P and Q are cyclic and let P = 〈x〉 and Q = 〈y〉 with |x| = pm
and |y| = qn. In this case, y−1xy = xr with rq
n ≡ 1 (mod pm), q | p − 1, 0 <
< r < pm and (pm, qn(r − 1)) = 1. Since 〈yq〉 is not S-quasinormal in G, we have
(yq)−1xyq = xr
q 6= x. So rq 6≡ 1 (mod pm). Since 〈yq2〉 is S-quasinormal in 〈x〉〈yq〉,
it follows from Lemma 3 that 〈yq2〉 is normal in 〈x〉〈yq〉. So (yq
2
)−1xyq
2
= xr
q2
= x.
Hence rq
2 ≡ 1 (mod pm) and y induces a power automorphism of order q2 in P. Surely,
yq induces a power automorphism of order q in P and every proper subgroup of 〈yq〉 is
normal in G. If xp 6= 1, then by Lemma 4, 〈xp〉〈yq〉 6= 〈xp〉 × 〈yq〉. Lemma 3 implies
that 〈yq〉 is normal in 〈xp〉〈y〉. Thus, 〈xp〉〈yq〉 = 〈xp〉 × 〈yq〉, a contradiction. So G is
of type (I).
(2) Assume that G is supersolvable and P is non-cyclic with d(P ) = k and Q = 〈y〉,
where d(P ) is the rank of P.
We can assume that
1 E . . . E R E P E . . . E G
is a chief series of G. By Maschke’s Theorem [10] (Theorem 8.1.2), there exists a
subgroup N of P such that P/Φ(P ) = R/Φ(P )×N/Φ(P ), where |N/Φ(P )| = p and
N/Φ(P ) E G/Φ(P ). Clearly, N E G, N 6≤ R and 1 E N E P E G is a normal series
of G. Applying Schreier’s Refinement Theorem [10] (Theorem 3.1.2), P has a maximal
subgroup K such that K is normal in G and K 6= R. Therefore, P has at least two
maximal subgroups R and K which are normal in G.
Now we prove k = 2. If k ≥ 3, then we can let P/Φ(P ) = 〈ā1〉 × 〈ā2〉 × . . .
. . . × 〈āk〉, where a1, a2, . . . , ak−1 ∈ R, a2, a3, . . . , ak ∈ K. Since R〈y〉 is an MSP -
group, it follows from Lemma 3 that (rΦ(R))
y
= rlΦ(R) for every r ∈ R, where l is
a positive integer. Thus (rΦ(P ))
y
= rlΦ(P ) for every r ∈ R. Similarly, (kΦ(P ))
y
=
= kmΦ(P ) for every k ∈ K, where m is a positive integer. It follows that al2Φ(P ) =
= (a2Φ(P ))
y
= am2 Φ(P ) and therefore l ≡ m (mod p). Hence (aiΦ(P ))
y
= aliΦ(P )
for i = 1, 2, . . . , k. It is easy to see that y induces a power automorphism in P/Φ(P ).
By Lemma 3, G is an MSP -group. This contradiction implies k = 2.
Now we let P/Φ(P ) = R/Φ(P )×K/Φ(P ) = 〈ā1〉 × 〈ā2〉 where a1 ∈ R, a2 ∈ K,
āy1 = āk11 and āy2 = āk22 . If k1 = k2, then G is an MSP -group by Lemma 3, a
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 9
1282 P. GUO, X. ZHANG
contradiction. Hence k1 6= k2. Furthermore, we have that P has only two maximal
subgroups which are normal in G. Clearly, at least one action of which y acts on
R and K is nontrivial. Without loss of generality, we may assume that y induces an
automorphism α of order q in R. Since every subgroup of R〈y〉 is an MSP -group and
by induction, it follows from Lemma 3 that every subgroup of R permutes with every
subgroup of 〈y〉. By Lemma 5, R is abelian and α is a power automorphism of order q in
R. By Lemma 4, α is fixed-point-free. So we have either K∩R = 1 if K〈y〉 = K×〈y〉
or K〈y〉 6= K × 〈y〉. If K ∩ R = 1 and K〈y〉 = K × 〈y〉, then P is an elementary
abelian group of order p2. If K〈y〉 6= K×〈y〉, similar arguments as above, K is abelian
and y induces a power automorphism of order q in K. Thus, Φ(P ) = R ∩K 6 Z(P ),
[R, 〈yq〉] = 1 and [K, 〈yq〉] = 1.
Hence 〈yq〉 E G. If |P : Z(P )| = p, then P is abelian. If |P : Z(P )| = p2,
then Φ(P ) = R ∩ K = Z(P ) and so P is minimal non-abelian. Clearly, all maximal
subgroups of G are only RQu, KQu and P 〈yq〉 where u ∈ G. Applying Agrawal’s
Theorem in [1], G is a minimal non-PST -group. An examination of the list of minimal
non-PST -groups in [11] (Theorem 1) , we have that G is only of type (II).
(3) Assume that P is cyclic and Q is non-cyclic. If Q has two non-cyclic maximal
subgroups Q1 and Q2, then Lemma 3, PQ1 = P × Q1 and PQ2 = P × Q2. Hence
Q = Q1Q2 is normal in G, a contradiction. Therefore, every maximal subgroup of Q is
cyclic or Q has a unique non-cyclic maximal subgroup.
Case 1. Every maximal subgroup of Q is cyclic. It is easy to see that Q is the
quaternion group Q8. Let Q = 〈a, b | a4 = 1, b2 = a2, b−1ab = a−1〉 and P = 〈z〉. If
zp 6= 1, then 〈zp〉Q = 〈zp〉 ×Q by Lemma 3. On the other hand, if 〈z〉〈a〉 = 〈z〉 × 〈a〉
and 〈z〉〈b〉 = 〈z〉 × 〈b〉, then G is nilpotent, a contradiction. Therefore, there exists
a nontrivial power automorphism that a or b acts on P by conjugation. Without loss
of generality, we assume that a acts on P by conjugation is nontrivial. By Lemma 4,
〈zp〉〈a〉 6= 〈zp〉 × 〈a〉. This contradicts 〈zp〉Q = 〈zp〉 ×Q. Thus, zp = 1. It follows that
〈a2〉 ≤ CG(P ) from P 〈a〉 is an MSP -group. If CG(P ) = P × 〈a2〉, then G/CG(P )
is an elementary abelian group of order 4. However, G/CG(P ) . Aut(P ) and Aut(P )
is cyclic, a contradiction. So there exists an element of order 4 contained in CG(P )
and CG(P ) = 〈x〉 is a cyclic group of order 4p where x is a generator of CG(P ).
Surely, G has an element y of order 4 such that y 6= xp. Now we let y−1xy = xr
where r 6≡ 1 (mod 4p). Since (y2)−1xy2 = xr
2
= x, we have r2 ≡ 1 (mod 4p). By
computations, we have G = 〈x, y | x4p = 1, y2 = x2p, y−1xy = x−1〉. So G is of
type (III).
Case 2. Let P = 〈x〉 and Q be Lemma 6 (I) with |Q| = qn. Namely, Q = 〈y, z |
yq
n−1
= zq = 1, [y, z] = 1〉 where n ≥ 3. Then Q has maximal subgroups H = 〈y〉,
K0 = 〈yq, z〉 and Ks = 〈yq, zys〉 = 〈zys〉 with s = 1, . . . , q−1, where K0 is the unique
non-cyclic maximal subgroup of Q. By Lemma 3, PK0 = P ×K0 and PH 6= P ×H.
For an MSP -group PH of G, we have that y induces an automorphism of order q in
P. Surely, z ∈ Z(G). Furthermore, similar arguments in Case 1, we can prove xp = 1.
Hence G = 〈x, y, z | xp = yq
n−1
= zq = 1, y−1xy = xr, [x, z] = 1, [y, z] = 1〉,
where r 6≡ 1 (mod p), rq ≡ 1 (mod p) and n ≥ 3. So G is of type (IV).
Case 3. Let P = 〈x〉 and Q be Lemma 6 (II) with |Q| = qn. Namely, Q =
= 〈y, z | yqn−1
= zq = 1, z−1yz = y1+p
n−2〉 where n ≥ 3 and n ≥ 4 if p = 2. In
the similar way as above, y induces a power automorphism of order q in P and 〈z〉 ≤
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 9
ON MINIMAL NON-MSP -GROUPS 1283
≤ CG(P ). Furthermore, we can also prove xp = 1, y−1xy = xr where r 6≡ 1 (mod p)
and rq ≡ 1 (mod p). So G is of type (V).
(4) Assume that G is minimal non-supersolvable and Q = 〈y〉 is cyclic.
Now we prove that: if Q ≤M lG, then Φ(P ) is a Sylow p-subgroup of M.
Denote M = P3Q, where P3 is a Sylow p-subgroup of M. By [P3, Q] ≤ P ∩P3Q =
= P3, we have NG(P3) ≥ P3Q = M. And since NP (P3) > P3, it follows that P3 is
normal in G. By Lemma 7 and the maximality of M, P3 = Φ(P ) is the Sylow p-
subgroup.
Case 1. If G is also a minimal non-nilpotent group, then by [17] (Theorem 2.8),
P is non-cyclic. Applying [5] (Theorem 3), G is of either type (VI) or type (VII).
Case 2. If G is not a minimal non-nilpotent group and P is abelian, applying
[4] (Theorems 9, 10), we assume that: G = PQ, where P = 〈a0, a1, . . . , aq−1〉 is an
elementary abelian p-group of order pq, Q = 〈y〉 is cyclic of order qr, qf is the highest
power of q dividing p − 1 and r > f ≥ 1. Define ayj = aj+1 for 0 ≤ j < q − 1 and
ayq−1 = ai0, where i is a primitive qf -th root of unity modulo p.
For an MSP -group P 〈yq〉 of G, by Lemma 3, yq induces a power automorphism
of order q on P. Hence ai
q
0 = ay
q2
0 = a0. Thus iq ≡ 1 (mod p) and f = 1. So G is of
type (VIII).
Case 3. Assume that G is not a minimal non-nilpotent group and P is non-abelian.
Applying [4] (Theorems 9, 10), we may assume that G = PQ such that P = 〈a0, a1〉
is an extraspecial group of order p3 with exponent p, Q = 〈y〉 is a cyclic group of order
2r with 2f the largest power of 2 dividing p − 1 and r > f ≥ 1, and ay0 = a1 and
ay1 = ai0x, where x ∈ 〈[a0, a1]〉 and i is a primitive 2f -th root of unity modulo p.
Since every subgroup of P 〈y2〉 is an MSP -group, by induction, it follows from
Lemma 3 that y2 induces a power automorphism of order dividing 2 in 〈a0〉. However
ay
2
0 = ay1 = ai0x 6∈ 〈a0〉, a contradiction. So G is not of the type as above.
Conversely, it is clear that a group of type (I) is a minimal non-MSP -group.
For type (II), we easily have that P has only two maximal subgroups 〈a〉 and 〈b〉
which are normal in G and P has a maximal subgroup H = 〈ab〉 such that HG = P.
So G is a minimal non-MSP -group.
For type (III), it is easy to see that 〈y〉 is not S-quasinormal in G, so G is not an
MSP -group. Since 〈x2p〉 = Z(G) is the unique subgroup of G of order 2 and G has
maximal groups of order 4p and 8, it follows that G is a minimal non-MSP -group.
For type (IV), G is not an MSP -group since Q has a maximal subgroup H =
= 〈y〉 is not S-quasinormal in G. Similar arguments in Case 2 of (3), G has maximal
subgroups PHu, PKu
0 , PK
u
s and Qu where u ∈ G, K0 = 〈yq, z〉, Ks = 〈zys〉 with
s = 1, . . . , q − 1. Since 〈yq〉 is normal in G and z ∈ Z(G), we have that all proper
subgroups of G are MSP -groups. So G is a minimal non-MSP -group.
For type (V), in the similar way as type (IV), we have that G is a minimal non-
MSP -group easily.
For types (VI) and (VII), it follows that G is non-supersolvable from [17] (Theo-
rem 2.8). By Lemma 1, G is not an MSP -group. So G is a minimal non-MSP -group.
For type (VIII), by Lemma 1, G is not an MSP -group. Similar arguments as above,
we have that G has maximal subgroups P 〈yq〉u and Qu where u ∈ G. It is easy to see
that G is a minimal non-MSP -group.
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1284 P. GUO, X. ZHANG
Acknowledgments. The authors express their deep sense of gratitude to Professor
X. Y. Guo, Department of Mathematics, Shanghai University, China for his work in
reviewing the paper and a number of valuable suggestions and comments.
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Received 12.03.11
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 9
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| id | umjimathkievua-article-2804 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:30:38Z |
| publishDate | 2011 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/b2/2da705b1a66ae5289af33ee7ee9a1fb2.pdf |
| spelling | umjimathkievua-article-28042020-03-18T19:36:55Z On minimal non- MSP -groups Про мiнiмальнi не MSP-групи Guo, P. Zhang, Xirong Го, П. Чжан, Хіронг A finite group $G$ is called an $MSP$-group if all maximal subgroups of the Sylow subgroups of $G$ are Squasinormal in $G$. In this paper, wc give a complete classification of those groups which are not $MSP$-groups but whose proper subgroups are all $MSP$-groups. Скiнченну групу $G$ називають $MSP$-групою, якщо всi максимальнi пiдгрупи силовських пiдгруп $G$ є $S$-квазiнормальними в $G$. Наведено повну класифiкацiю груп, якi не є $MSP$-групами, але всi їх власнi пiдгрупи є $MSP$-групами. Institute of Mathematics, NAS of Ukraine 2011-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2804 Ukrains’kyi Matematychnyi Zhurnal; Vol. 63 No. 9 (2011); 1279-1278 Український математичний журнал; Том 63 № 9 (2011); 1279-1278 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2804/2361 https://umj.imath.kiev.ua/index.php/umj/article/view/2804/2362 Copyright (c) 2011 Guo P.; Zhang Xirong |
| spellingShingle | Guo, P. Zhang, Xirong Го, П. Чжан, Хіронг On minimal non- MSP -groups |
| title | On minimal non- MSP -groups |
| title_alt | Про мiнiмальнi не MSP-групи |
| title_full | On minimal non- MSP -groups |
| title_fullStr | On minimal non- MSP -groups |
| title_full_unstemmed | On minimal non- MSP -groups |
| title_short | On minimal non- MSP -groups |
| title_sort | on minimal non- msp -groups |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2804 |
| work_keys_str_mv | AT guop onminimalnonmspgroups AT zhangxirong onminimalnonmspgroups AT gop onminimalnonmspgroups AT čžanhírong onminimalnonmspgroups AT guop prominimalʹninemspgrupi AT zhangxirong prominimalʹninemspgrupi AT gop prominimalʹninemspgrupi AT čžanhírong prominimalʹninemspgrupi |