Ukrainian Mathematical Journal, Vol. 61, No. 12, 2009

STRUCTURE OF FINITE GROUPS WITH S -QUASINORMAL THIRD MAXIMAL SUBGROUPS Yu. V. Lutsenko and A. N. Skiba

UDC 512.542

We study finite groups whose 3-maximal subgroups are permutable with all Sylow subgroups.

1. Introduction All groups considered in the present paper are finite. Recall that a subgroup H of a group G is called a 2-maximal subgroup (or a second maximal subgroup) of the group G if H is a maximal subgroup in a certain maximal subgroup M of the group G: By analogy, one can define 3-maximal subgroups, 4-maximal subgroups, etc. It is easy to see that, in nonsupersolvable groups, a subgroup can be both n-maximal and m-maximal simultaneously for n ¤ m: In this connection, we say that a subgroup H of a group G is a strictly n-maximal subgroup in G if H is an n-maximal subgroup in G but is not an n-maximal subgroup in any proper subgroup of G: For example, in the group SL.2; 3/; the unique subgroup of order 2 is 2-maximal but not strictly 2-maximal. The relationship between n-maximal semigroups .n > 1/ of a group G and the structure of the group G was studied by many authors. Apparently, the first result in this direction was obtained by Huppert [1], who proved that a group is supersolvable if all 2-maximal subgroups of it are normal. In the same paper, it was also proved that if each third maximal subgroup of a group G is normal in G; then the commutant G 0 of the group G is nilpotent, and the order of each principal factor of the group G is divided by at most two (not necessarily different) prime numbers. Huppert’s work [1] stimulated numerous investigations in this direction. In particular, developing Huppert’s results, Janko obtained a description of groups in which 4-maximal subgroups are normal. He proved that if each 4-maximal subgroup of a solvable group G is normal in G and the order of G is divided by at most four different prime numbers, then G is a supersolvable group [2]. A year later, Janko studied groups that do not contain 5maximal subgroups other than the identity one [3]. Among the early works in this direction, we also note Agrawal’s work [4], where it was proved that a group is supersolvable if each 2-maximal subgroup of it is S -quasinormal (a subgroup H of a group G is called S -quasinormal, or S -permutable, in G if H is permutable with all Sylow subgroups of G). The results of Huppert and Janko were also naturally developed by Mann [5], who analyzed the structure of groups in which each n-maximal subgroup is subnormal. Later, Asaad [6] improved the results of Huppert and Janko for strictly n-maximal subgroups with n D 2; 3; 4: In the present paper, we give a complete description of groups all 3-maximal subgroups of which are S quasinormal. Based on this result, we also solve, in the class of nonnilpotent groups, the Huppert problem of complete description of groups in which 3-maximal subgroups are normal. Gomel University, Gomel, Belarus. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 12, pp. 1630–1639, December, 2009. Original article submitted February 10, 2009; revision submitted June 15, 2009. 0041–5995/09/6112–1915

c 2009 Springer Science+Business Media, Inc.

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Y U. V. L UTSENKO AND A. N. S KIBA

1916

2. Preliminary Results Recall some properties of Schmidt groups necessary for what follows (see [7], Chap. VI). Lemma 2.1. If G is a Schmidt group, then the following assertions are true: (1) G D ŒP hai; where P is the Sylow p-subgroup of the group G and hai is the Sylow q-subgroup of the group GI (2) G has exactly two classes of maximal subgroups, whose representatives are the subgroups P haq i and P 0 haiI (3) if P is non-Abelian, then its center, commutant, and Frattini subgroup coincide and have either the exponent p or the exponent 4 (if p D 2). We present Theorem 2.1 from [8] in the form of a lemma. Lemma 2.2. Let G be a nonnilpotent group. Then the following conditions are equivalent: (1) G is a supersolvable Schmidt group; (2) each 2-maximal subgroup of the group G is normal; (3) each strictly 2-maximal subgroup of the group G is S -quasinormal. The following statement was proved in [9, 10]: Lemma 2.3. If each second maximal subgroup of an unsolvable group G is nilpotent, then G is isomorphic to one of the groups A5 and SL.2; 5/: Recall some properties of S -quasinormal subgroups. Lemma 2.4 [11]. Let G be a group and let H  K  G: Then the following assertions are true: (1) if H is S -permutable in G; then H is S -permutable in KI (2) if H is normal in G; then K=H is S -permutable in G=H if and only if K is S -permutable in GI (3) if H is S -permutable in G; then H is subnormal in G: The following lemma can be proved by induction on n: Lemma 2.5. Let E be an n-maximal subgroup of a q-nilpotent group G: Then jG W Ej D q ˛ s; where ˛  n and .s; q/ D 1: Lemma 2.6. Let T be a subgroup of a group G and let Q D hai be a cyclic q-subgroup of the group G: If T \ Qx D 1 for any x 2 G and the group G is q-nilpotent, then jQj divides jG W T j: Proof. Assume that T ¤ G: By induction on jG W T j; we show that jQj divides jG W T j: Let T  M; where M is a maximal subgroup in G: Assume that haix  M for a certain x 2 G: Since .haix /m \ T D 1 for all m 2 M; we can conclude by induction that jhaix j D jhaij divides jM W T j; and, hence, jQj D jhaij divides jG W T j: Thus, haix — M for all x 2 G: Then M is normal in G: Since .hai \ M /m \ T D 1 for all m 2 M; we can conclude by induction that jhai \ M j divides jM W T j: This implies that jhai \ M j divides jG W T j: Since G=M D haiM=M ' hai=M \ hai; we conclude that jhai W M \ haij divides jG W T j: Therefore, jQj D jhai W M \ haijjhai \ M j divides jG W T j: The lemma is proved.

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3. Structure of Groups All 3-Maximal Subgroups of Which Are S -Quasinormal In what follows, p; q; and r are prime (not necessarily different) numbers. In the theorems presented below, P; Q; and R denote a Sylow p-subgroup, a Sylow q-subgroup, and a Sylow r-subgroup, respectively, of the group G: ˝ ˇ 1 ˇ 2˛ Let Mˇ .q/ denote the q-group a; b j aq D b q D 1; ab D a1Cq ; where ˇ > 2; and ˇ > 3 if q D 2 (see [12, p. 190]). Theorem 3.1. Each 3-maximal subgroup of a group G is S -quasinormal in G if and only if either the group G is nilpotent, or jGj D p ˛ q ˇ r ; where ˛ C ˇ C  3; or G is isomorphic to SL.2; 3/; or G is a supersolvable group of one of the following types: (1) G is a Schmidt group; (2) G D ŒP Q; where jP j D pI jQj D q ˇ .ˇ  3/I the group Q is either a cyclic group, or an Abelian group of the type .q ˇ 1 ; q/; or a group isomorphic either to a quaternion group of order 8 or to the group Mˇ .q/I and CQ .P / D ˇ 2 .Q/I (3) G D ŒP Q; where P is a cyclic group of order p 2 ; both groups ˆ.P /Q and G=ˆ.P / are Schmidt groups, and the maximal subgroup from Q coincides with Z.G/I (4) G D ŒP1  P2 Q; where jP1 j D jP2 j D p; P1 Q is a Schmidt group, and the group P2 Q is either a nilpotent group or a Schmidt group; (5) G D .ŒP Q/R; where P and R are minimal normal subgroups of the group G; jP j D p; jRj D r; Q is a cyclic group, and F .G/ D PRˆ.Q/: Proof. Necessity. Let G be a nonnilpotent group each 3-maximal subgroup of which is S -quasinormal. Then, by virtue of Lemma 2.4(3), each 3-maximal subgroup of G is subnormal. By virtue of Lemmas 2.4(1) and 2.2, each maximal subgroup of the group G is either a nilpotent group or a supersolvable Schmidt group. Thus, G is a group each 2-maximal subgroup of which is nilpotent. First, assume that the group G is unsolvable. According to Lemma 2.3, G is isomorphic either to the group A5 or to the group SL.2; 5/: Since the group A5 ' SL.2; 5/=Z.SL.2; 5// contains a nontrivial 3-maximal subgroup, this case is impossible by virtue of assertions (2) and (3) of Lemma 2.4. Thus, G is a solvable group, and each proper subgroup H of G is either a nilpotent group or a Schmidt group. Furthermore, if H is a Schmidt group, then H is maximal in G: Since, the index of any maximal subgroup in a solvable group is the power of a prime number, and the number of different prime divisors of a Schmidt group is equal to 2; we conclude that .G/  3: In what follows, we assume that jGj D p ˛ q ˇ r ; where ˛ C ˇ C > 3: I. First, assume that G D ŒP Q is a Schmidt group. Let P be an Abelian group and let jQj > q: In this case, the group G has a 3-maximal subgroup P1 Q2 ; where P1 is a certain maximal subgroup in P and Q2 is a 2-maximal subgroup in Q: By assumption, we have .P1 Q2 /Q D Q.P1 Q2 /: Therefore, P1 Q is a subgroup of G: By virtue of Lemma 2.1(2), Q is a maximal subgroup of G: Therefore, P1 D 1: Thus, jP j D p; and G is a supersolvable group of type 1. Now let jQj D q: In this case, each 2-maximal subgroup P2 of P is a 3-maximal subgroup of G: Then, by assumption, we have P2 Q D QP2 ; and, hence, P2 Q is a subgroup of G: Using the maximality of Q in G; we get P2 D 1: Therefore, jP j D p 2 and, hence, jGj D p 2 q; which contradicts the original assumptions concerning the group G:

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Now let P be a non-Abelian group and let jQj > q: In this case, the group G has a 3-maximal subgroup P1 Q2 ; where P1 is a certain maximal subgroup of P and Q2 is a 2-maximal subgroup of Q: By assumption, we have .P1 Q2 /Q D Q.P1 Q2 /; and, hence, P1 Q is a subgroup of G: By virtue of assertions (2) and (3) of Lemma 2.1, ˆ.P /Q is a maximal subgroup of G: Therefore, P1 D ˆ.P /: Thus, P is a cyclic group, which contradicts the statement that P is a non-Abelian group. Now let jQj D q: It is clear that each 2-maximal subgroup P2 of P is a 3-maximal subgroup of G: Then, by assumption, we have P2 Q D QP2 : Hence, P2 Q is a subgroup of G: Therefore, P2 ˆ.P /Q is a subgroup of G; and, hence, by virtue of the maximality of ˆ.P /Q in G; we have either P2 ˆ.P /Q D ˆ.P /Q or P2 ˆ.P /Q D G: It is clear that the second case is impossible. Therefore, P2 ˆ.P /Q D ˆ.P /Q; whence P2  ˆ.P /: If P2 < ˆ.P /; then P is an Abelian group, which contradicts the case considered. Therefore, P2 D ˆ.P /; and, hence, ˆ.P / is the unique 2-maximal subgroup of P: Since the group P is not cyclic, by virtue of Theorems 8.2 and 8.4 in [13] (Chap. III) we can conclude that jˆ.P /j D 2 and P is isomorphic to a quaternion group Q8 of order 8: Since jP =ˆ.P /j D 4; we have 4  1 .mod q/: This implies that q D 3; and, hence, the group G is isomorphic to the group SL.2; 3/: II. Assume that G is not a Schmidt group and .G/ D fp; qg: First, assume that the group G has a normal Sylow subgroup P; i.e., G D ŒP Q: Also assume that the group G has a pair of Schmidt subgroups of the form A D ŒP Q1 and B D ŒP1 Q .P1 < P; Q1 < Q/: Then A and B are maximal subgroups of the group G: Each 2-maximal subgroup in the group A is S -quasinormal. Therefore, according to Lemma 2.2, we have jP j D p: By analogy, we can show that jP1 j D p; which is impossible. Thus, either all Schmidt subgroups of the group G contain a Sylow p-subgroup of G or all these subgroups contain a Sylow q-subgroup of G: If all Schmidt subgroups of the group G contain P; then jP j D p and jQj D q ˇ ; ˇ  3: Let x be an arbitrary element of Q such that jxj D q ˇ 2 : Since jQj D q ˇ ; we conclude that x is contained in a certain maximal subgroup Q1 of the group Q: It is clear that PQ1 is a maximal subgroup of G; and, hence, either PQ1 is a nilpotent group or PQ1 is a Schmidt group with jP j D p and jQ1 j D q ˇ 1 : It is obvious that, in both cases, the element x centralizes the group P; whence CQ .P / D ˇ 2 .Q/: Let Q be an Abelian group. If, in addition, the group Q is cyclic, then G is a supersolvable group of type 2. Let Q be a noncyclic group and let H be a Schmidt subgroup of the group G: Then jG W H j D q and H D ŒP Q1 ; where Q1 is a cyclic maximal subgroup of Q: According to the main theorem on finite Abelian groups, we have Q D Q1  Cq ; where jCq j D q: It is clear that, in this case, G is a supersolvable group of type 2. Now assume that Q is a non-Abelian group. Then Q has two different maximal subgroups Q1 and Q2 : Since the group G is not nilpotent, at least one of the subgroups PQ1 and PQ2 is not nilpotent. Therefore, this subgroup is a Schmidt group. This means that all maximal subgroups of Q but one are cyclic. Now assume that q D 2 and ˇ D 3: Then, according to Theorem 4.4 in [12] (Chap. V), Q is isomorphic either to the group Q8 or to a dihedral group. In the latter case, we have Q D Œhaihbi; where jaj D 22 ; jbj D 2; and ab D a 1 : Then Q has exactly three maximal subgroups, namely hai; ha2 ihbi; and ha2 ihabi: Therefore, subgroups of the form ha2 i; hbi; and habi are 3-maximal subgroups of G: It is known that a dihedral group D possesses the property 1 .D/ D D (see [12], Chap. V, Theorem 4.3). This means that one of the subgroups of order 2 of the dihedral group, say, habi; is not contained in QG : Therefore, this subgroup is not S -quasinormal in G but is a 3-maximal subgroup of G: The contradiction obtained shows that Q ' Q8 : Thus, G is a supersolvable group of type 2. Now assume that either q D 2 and ˇ > 3 or q is an odd prime number. Then, in the first case, according to Theorem 4.4 in [12] (Chap. V), Q is isomorphic to one of the groups Mˇ .2/; Dˇ ; Qˇ ; and Sˇ : In the second case, it is isomorphic to the group Mˇ .q/ (see [12, pp. 190, 191]). If Q is isomorphic to one of the groups Dˇ ; Qˇ ; and Sˇ ; then, according to Theorem 4.3 in [12] (Chap. V), the quotient group Q=Z.Q/ is isomorphic to

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Dˇ 1 : Since the group Dˇ 1 contains two noncyclic maximal subgroups, we conclude that the group Q has at least two noncyclic maximal subgroups, which contradicts the statement proved above. Therefore, the group Q cannot be isomorphic to one of the groups Dˇ ; Qˇ ; and Sˇ : Thus, the group Q is isomorphic to Mˇ .q/; and, consequently, G is a supersolvable group of type 2. Now assume that any Schmidt subgroup of the group G contains a certain Sylow q-subgroup of G: This means that Q D hai is a cyclic group and P haq i is a maximal subgroup of the group G with index q: Since the subgroup P haq i is not a Schmidt group, it is nilpotent. Therefore, P haq i D P haq i D F .G/; and, hence, haq i is contained in Z.G/: In this case, either each maximal subgroup of the group G containing a Sylow q-subgroup of the group G is a Schmidt group or G has a nilpotent maximal subgroup that contains a Sylow q-subgroup of the group G: Assume that the first case takes place. Let M be an arbitrary maximal subgroup of the group G of the form P1 Q; where P1 < P: Then M is a Schmidt group with jP1 j D p: First, assume that P is a non-Abelian group. It is easy to show that, in this case, P1 D ˆ.P / and ˆ.P / is the unique 2-maximal subgroup in P: Therefore, P is not a cyclic group in the case considered. According to Theorems 8.2 and 8.4 in [13] (Chap. III), jˆ.P /j D 2 and P is isomorphic to the group Q8 : However, ˆ.P /Q D Œˆ.P /Q is a Schmidt group, which is impossible. Now let P be an Abelian group. Assume that ˆ.P / ¤ 1: Then P1 D ˆ.P / is a normal subgroup of order p of the group G: Assume that jQj > q: Let P0 be a certain maximal subgroup of the group P and let Q2 be a 2-maximal subgroup of the group Q: Then P0 Q2 is a 3-maximal subgroup of G: By assumption, we have .P0 Q2 /Q D Q.P0 Q2 /; and, therefore, P0 Q is a subgroup of the group G: Since the subgroup ˆ.P /Q is maximal in the group G; we get ˆ.P /Q D P1 Q: This means that ˆ.P / is a maximal subgroup of P: Therefore, P is a cyclic group of order p 2 : Since ˆ.P /  ˆ.G/ and the group G is not nilpotent, we conclude that G=ˆ.P / is a supersolvable Schmidt group. Since each maximal subgroup of the group G that contains a Sylow q-subgroup of the group G is a supersolvable Schmidt group, we conclude that the maximal subgroup of the group Q coincides with Z.G/: Therefore, G is a supersolvable group of type 3. Now assume that jQj D q: Let P2 be an arbitrary 2-maximal subgroup of P: Then P2 is a 3-maximal subgroup of G: Therefore, P2 Q is a subgroup of the group G: Reasoning as above, we show that ˆ.P / is the unique 2-maximal subgroup of P: Since P is Abelian, it is a cyclic group. This, in turn, implies that jGj D p 2 q; which contradicts the original assumption concerning the group G: Assume that ˆ.P / D 1: In this case, P is an elementary p-group. Consider the maximal subgroup T of the group G such that G D ŒP1 T: Since T D ŒP2 Q is a Schmidt group in the case considered, by virtue of Lemma 2.2 we can conclude that P2 is a normal subgroup of order p in the group G: Thus, G is a supersolvable group of type 4. Now assume that G has a nilpotent maximal subgroup M that contains the subgroup Q: Then the group M has the form P1  Q; where P1 < P: Assume that G does not have maximal subgroups that are Schmidt groups. Then each maximal subgroup of the group G is nilpotent, and, hence, G is a Schmidt group, which contradicts the conditions of the case considered. Therefore, among the maximal subgroups of the group G; there is a subgroup H that is a Schmidt group. Without loss of generality, we can assume that Q  H: By virtue of Lemma 3.9 in [13] (Chap. II), we have HM D G: Let Hp be a Sylow p-subgroup of the group H: The subgroup Hp is not contained in P1 : Therefore, Hp \ P1 D 1; whence H \ M D Q: Assume that jP1 j > p: Let E be a 2-maximal subgroup of M whose index is equal to p 2 : Then E is a 3-maximal subgroup of G and Q  E: It follows from the conditions of the theorem that EQx is a subgroup of G for all x 2 G: According to Lemma 4.7 in [13] (Chap. VI), QQx is a Sylow q-subgroup of EQx : Therefore, the subgroup Q D Qx is normal in the group G: The contradiction obtained shows that jP1 j D p: It is clear that Hp and P1 are normal subgroups of order p in the group G: Therefore, G is a supersolvable group of type 4.

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Now assume that the group G does not have normal Sylow subgroups. Let H be a normal subgroup of the group G whose index is equal to p: Then Q is a subgroup of H: Assume that the subgroup Q is normal in the group H: Then the subgroup Q is normal in the group G; which contradicts the conditions of the case considered. Therefore, H D ŒP1 Q is a Schmidt subgroup of the group G; and, hence, Q is a cyclic group. According to Lemma 2.2, P1 is a normal subgroup of order p in the group G: Let NG .Q/ be a nilpotent subgroup of the group G: Then Q  Z.NG .Q// because Q is an Abelian group. In this case, according to Theorem 14.3.1 in [14], the group G has a normal q-complement, which contradicts the conditions of the case considered. Therefore, NG .Q/ D ŒQhbi is a Schmidt subgroup of G and jQj D q: This yields jGj D p 2 q; which contradicts the original assumption concerning the group G: III. Finally, we consider the case where .G/ D fp; q; rg; where p; q; and r are different prime divisors of jGj: Let M denote a normal subgroup of the group G such that jG W M j D q: Then M is either a nilpotent group or a Schmidt group. Assume that M is a nilpotent group. Then G D ŒP  RQ and M D P  R  Q1 ; where Q1 is a certain maximal subgroup of Q: The subgroups PQ and RQ cannot be simultaneously nilpotent. Therefore, either PQ and RQ are Schmidt groups or one of these subgroups, say, RQ; is nilpotent, and the other is a Schmidt group. Assume that the first case takes place. Then the subgroups PQ and RQ are maximal in G: Therefore, by virtue of Lemma 2.2, we have jP j D p and jRj D r: Furthermore, according to Lemma 2.1(1), Q D hai is a cyclic group and haq i is a subgroup of Z.hPQ; RQi/ D Z.G/: Now assume that the subgroup PQ D P hai is a Schmidt group and the subgroup RQ is nilpotent. Then the subgroup PQ is maximal in G; whence G D PQ  R; where jRj D r: According to Lemma 2.2, P is a local subgroup of order p in G: Since haq i is a characteristic subgroup of Q and Q is normal in RQ; we conclude that the subgroup haq i is normal in RQ: Since haq i is also normal in the group PQ; we establish that the subgroup haq i is normal in G: Thus, G is a group of type 5. Now assume that M is a Schmidt group and G is not a group of type 5. Without loss of generality, we can assume that M D ŒRP; where P D hbi is a cyclic group. Then G D ŒM Q D ŒŒRP Q; where Q is a group of prime order q that is not a normal subgroup of G: Indeed, if Q is a normal subgroup of G; then G D M  Q is a group of type 5. Since M D ŒRP is a Schmidt group, it follows from Lemma 2.2 that R is a normal subgroup of order r in the group M and, hence, in the group G: Let RQ be a nilpotent group. If PQ is also a nilpotent group, then the subgroup Q is normal in G; which contradicts the conditions of the case considered. Therefore, PQ D ŒP Q is a Schmidt group with jP j D p: Since CG .R/ D RQ; we can conclude that Q is normal in G; which again contradicts the conditions of the case considered. Thus, RQ D ŒRQ is a Schmidt group. Now assume that the subgroup PQ D ŒP Q is a Schmidt group. Since jP j D p; we have p 1 D q˛ for a certain natural ˛: By analogy, taking into account that RQ and RP are Schmidt groups and jRj D r; we establish that r 1 D qˇ and r 1 D p for certain natural ˇ and : Therefore, p D qˇ 1 D 1 C q˛; which is impossible. Thus, PQ is a nilpotent group and, hence, G D ŒR.P  Q/I furthermore, the maximality of the subgroup RQ in G implies that P D hbi is a group of prime order p: Therefore, R D F .G/ and jGj D pqr; which contradicts the original assumption concerning the group G: Sufficiency. Note that each subgroup of a nilpotent group is S -quasinormal, and if jGj D p ˛ q ˇ r ; where p; q; and r are prime (not necessarily different) numbers and ˛ C ˇ C  3; then either G does not contain 3-maximal subgroups or 1 is the only 3-maximal subgroup in G: Moreover, if either G is isomorphic to the group SL.2; 3/ or G is a supersolvable group of one of types 1, 3–5, then the direct verification shows that all 3-maximal subgroups of G are normal. Thus, in the proof of the sufficiency part of Theorem 3.1, it suffices to

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consider the case where G is a supersolvable group of type 2. Furthermore, if Q is either a cyclic group, or an Abelian group of the type .q ˇ 1 ; q/; or a group isomorphic to a quaternion group of order 8, then one can directly verify that all 3-maximal subgroups of G are normal. Assume that Q ' Mˇ .q/: Then it follows from the conditions and the definition of the group Mˇ .q/ that G D .ŒP Q1 /Cq ; where jP j D p; CQ .P / D ˇ 2 .Q/; Q1 D hai; Cq D hbi; jQ1 Cq j D q ˇ ; jaj D q ˇ 1 ; ˇ 2 ab D a1Cq ; ˇ  3 for odd q; and ˇ > 3 for q D 2: Let T be an arbitrary 3-maximal subgroup of G: We show that T is S -quasinormal in G: First, assume that T \ Q1x ¤ 1 for a certain x 2 G: Then Q1x has a proper subgroup Z such that Z  T and jZj D q: By assumption, we have Z  CG .P /; and, hence, Z is normal in G: Then G=Z D ŒP Z=Z.Q1 Cq =Z/; where jQ1 Cq =Zj D q ˛ ; ˛  2 for odd q; and ˛ > 2 for q D 2: According to Theorem 4.3 in [12] (Chap. V), we have Z D Q0 ; and, hence, Q1 Cq =Z is an Abelian group of the type .q ˛ 1 ; q/: If ˛ > 2; then, as indicated above, T =Z is an S -quasinormal 3-maximal subgroup of G=Z: Then, by virtue of Lemma 2.4(2), T is S -quasinormal in G: If ˛ D 2; then, obviously, jGj D pq 3 : Since Z  T and T is a 3-maximal subgroup of G; we conclude that Z D T is normal in G: Now assume that T \ Q1x D 1 for all x 2 G: Since the group G is q-nilpotent, it follows from Lemma 2.6 that jQ1 j divides jG W T j; and, according to Lemma 2.5, we have either jG W T j D q 3 or jG W T j D pq 2 : Assume that the first case takes place. Then P  T; and, hence, T =P is an S-quasinormal subgroup of G=P by virtue of the nilpotency of G=P: Therefore, according to Lemma 2.4(2), T is S-quasinormal in G: Now assume that jG W T j D pq 2 : In this case, jQ1 j divides jG W T j: Therefore, T ' Cq : Using the condition CQ .P / D ˇ 2 .Q/; one can easily show that, in this case, T is S -quasinormal in G: Thus, each 3-maximal subgroup of the group G is S-quasinormal. The theorem is proved. Corollary 3.1. Suppose that each third maximal subgroup of a group G is S -quasinormal in G: If j.G/j  3; then G is supersolvable. If j.G/j  4; then G is nilpotent. 4. Solution of the Huppert Problem in the Nonnilpotent Case In this section, we use Theorem 3.1 for the solution, in the nonnilpotent case, of the Huppert problem [1] of complete description of groups all third maximal subgroups of which are normal. Theorem 4.1. Let G be a nonnilpotent group. Each 3-maximal subgroup of the group G is normal in G if and only if either jGj D p ˛ q ˇ r ; ˛ C ˇ C  3; or G is isomorphic to SL.2; 3/; or G is a supersolvable group of one of the following types: (1) G is a Schmidt group; (2) G D ŒP Q; where jP j D pI jQj D q ˇ .ˇ  3/I Q is either a cyclic group, or an Abelian group of the type .q ˇ 1 ; q/; or a group isomorphic to a quaternion group of order 8; or a group isomorphic to the group Mˇ .q/ .ˇ > 4/I and CQ .P / D ˇ 2 .Q/I (3) G D ŒP Q; where P is a cyclic group of order p 2 ; both ˆ.P /Q and G=ˆ.P / are Schmidt groups, and the maximal subgroup of Q coincides with Z.G/I (4) G D ŒP1  P2 Q; where jP1 j D jP2 j D p; P1 Q is a Schmidt group, and P2 Q is either a nilpotent group or a Schmidt group; (5) G D .ŒP Q/R; where P and R are minimal normal subgroups of the group G; jP j D p; jRj D r; Q is a cyclic group, and F .G/ D PRˆ.Q/:

Y U. V. L UTSENKO AND A. N. S KIBA

1922

Proof. Necessity. Let G be a nonnilpotent group each 3-maximal subgroup of which is normal. Then each 3-maximal subgroup of the group G is obviously S -quasinormal. Therefore, G is a group of one of the types described in Theorem 3.1. Let G be a supersolvable group of type 2 from Theorem 3.1 and let Q ' Mˇ .q/; where ˇ  3 for odd q and ˇ > 3 for q D 2: It follows from the conditions and the definition of the group Mˇ .q/ that G D .ŒP Q1 /Cq ; where jP j D p; CQ .P / D ˇ 2 .Q/; Q1 D hai; Cq D hbi; jQ1 Cq j D q ˇ ; jaj D q ˇ 1 ; ˇ 2 and ab D a1Cq : If ˇ D 3; then, by assumption, the subgroup Cq is a normal 3-maximal subgroup of G; which contradicts the structure of the group Mˇ .q/: If ˇ D 4; then, by assumption, P Cq is a normal 3-maximal subgroup of G: Since CQ .P / D ˇ 2 .Q/ and jCq j D q; we conclude that P Cq is nilpotent. This, in turn, yields the normality of the subgroup Cq in G; which again contradicts the structure of the group Mˇ .q/: Therefore, in the case where G is a supersolvable group of type 2 from Theorem 3.1 and Q ' Mˇ .q/; we have ˇ > 4: Thus, G is a group of one of the types described in Theorem 4.1. Sufficiency. The sufficiency is verified in the same way as in the proof of Theorem 3.1. The theorem is proved. Corollary 4.1. Suppose that each third maximal subgroup of a group G is normal in G: If j.G/j  3; then G is supersolvable. If j.G/j  4; then G is nilpotent. Corollary 4.2 [1]. Suppose that each third maximal subgroup of a group G is normal in G: Then the commutant G 0 of the group G is nilpotent, and the order of each principal factor of the group G is not divided by p 3 for all prime p: The example presented below shows that, in the general case, the class of groups with S -quasinormal third maximal subgroups is broader than the class of groups all third maximal subgroups of which are normal. Example 4.1. Let Q D hx; y j x 9 D y 3 D 1; x y D x 4 i and let Z7 be a group of order 7: Then 1 .Q/ D  is an Abelian group of order 9: Since the quotient group Q= is isomorphic to a third-order subgroup of the automorphism group Aut .Z7 /; we can construct the group G D ŒZ7 Q: It is clear that hyi is a 3-maximal subgroup of G; hyi is nonnormal in G; and G is a group each 3-maximal subgroup of which is S-quasinormal. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

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