The structure of finite groups and ɵ-pairs of general subgroups

Abstract Using the concept of ɵ-pairs of proper subgroups of a finite group, we obtain some critical conditions of the supersolvability and nilpotency of finite groups.


Introduction
In this paper, all groups considered are finite and G stands for a finite group. Let .G/ stand for the set of all prime divisors of jGj. We use "M É G" to denote that M is a maximal subgroup of G; We write "N C har G" to mean that N is a characteristic subgroup of G. The other notations and terminologies are standard (see [4]).
It is well-know that each maximal subgroup of a solvable group is complement of a chief factor of G. Taking this elementary fact as starting point, N. P. Mukherjee and P. Bhattacharya [7] introduced the concept of Â -pairs of maximal subgroups and obtained some conditions for solvability, supersolvability or nilpotency of finite groups by using the properties of Â -pairs of maximal subgroups of G. Since then, many interesting results have been subsequently obtained (see [1][2][3]). Following the idea of Mukherjee and Bhattacharya, X. H. Li and S. H. Li [5] introduced the concept of Â -pairs of general subgroups. Let H be a subgroup of G. We call .A; B/ a Â -pair of H in G if .i/ A Ä G, hH; Ai D G and B D .A \ H / G ; .i i / If A 1 =B is a proper subgroup of A=B and A 1 =B C G=B, then G ¤ hH; A 1 i. If A C G, then .A; B/ is called a normal Â -pair of H in G. They gave sufficient conditions of supersolvability and nilpotence of a finite group by using the concept of Â -pair of some maximal subgroups of Sylow subgroups of G.
Given H < G, we denote by Â.H / the family of all Â -pairs of H . A partial ordering Ä on Â.H / is introduced as follows: We call a maximal element with respect to Ä a maximal Â -pair (see [5]). In [10] and [11], Wang introduced the concept of c-normal subgroups (c-supplemented Some authors have investigated the structure of a finite group G under the assumption that some subgroups of prime power order of G have those generalized normality in G, and obtained many results (see [8,10] and [11]). By [5,Theorem 1], the definitions of c-normality etc. are developed by imposing some conditions on Â -pairs. Hence, all results, obtainable by using a c-normal subgroup and c-supplemented subgroup of G, have analogs in the Â -pairs of subgroups of G. Based on the fact that the general subgroup H of G has a maximal Â -pair, then there exists a normal Â -pair which satisfies some conditions (see Lemma 3 (2) in [5] of subgroups. In this paper, we investigate the supersolvability and nilpotency of a group by using the concept of normal Â -pair of subgroups with order p k of a Sylow p-subgroup of a normal subgroup N of G for a given positive integer k. In comparison to the assumptions of [8, Theorem 0.1], we consider all cyclic subgroups of order 2 and not all cyclic subgroups of order 2 and 4 when p D 2. And we generalize the results in [6].

Preliminary results
. Let G be a group, p, q be different prime divisors of jGj, P a non-cyclic Sylow p-subgroup of G and Q a Sylow q-subgroup of G. If any maximal subgroup of P (except one) has a q-closed supplement in G, then Q is normal in G.
Lemma 2.6. Assume that every maximal subgroup of Sylow subgroup P of G has a nilpotent supplement in G, then G is nilpotent.
Proof. By [9, Theorem 1.4], G is supersolvable. Let q D max .G/, then Q C G, where Q 2 Syl q .G/. It is clear that G=ˆ.Q/ satisfies the hypothesis, thenˆ.Q/ D 1 by the formation of nilpotent groups is saturated, so Q is an elementary abelian q-subgroup. Since G=Q is nilpotent, we have G=Q D P 1 Q=Q P 2 Q=Q If Q is non-cyclic, then P i C har P i Q by Lemma 2.5. We conclude that P i C har P i Q C G, so P i C G, hence G is nilpotent.

Main results
Theorem 3.1. Assume that every Sylow subgroup P of a normal subgroup N of G has a subgroup U with 1 < jU j < jP j such that every subgroup H of P of order jU j has a normal Â -pair .C; D/ with C =D supersolvable, then G is supersolvable.
Proof. Assume that the result is false and let G be a counterexample with least jGj C jN j. By Lemma 2.4, G is solvable.
Step 1. If L is a minimal normal subgroup of G contained in P 2 Syl p .N /, then jLj Ä jU j.
Suppose that jLj > jU j. Then every subgroup H of L of order jU j has a normal Â -pair .C; D/ such that C =D is supersolvable. Thus G D H C and D D .H \ C / G . Since D Ä H < L, we have D D 1 by the minimal normality of L in G. Obviously, C \ L C G, then C \ L D 1 or C \ L D L. If the former case is held, then C \ H D 1, so jGj D jCH j D jCLj, that is, jH j D jLj, a contradiction. Thus C \ L D L, L Ä C , hence G D CH D CL D C . By the hypothesis, C =D D G is supersolvable, a contradiction.
Step 2. For every minimal normal subgroup L of G contained in N , the factor group G WD G=L is supersolvable.
Let p be a prime and jLj D p˛. Let Q be a q-Sylow subgroup of N . Assume that jLj D jU j. By the hypothesis, there is a normal Â -pair .C; D/ for L such that C =D is supersolvable, then G D LC and D D .L \ C / G D L \ C . By the minimal normality of L in G, we have L \ C D L or L \ C D 1. If the former case is true, then L Ä C , that is, G D C and C =D D G=L is supersolvable. If the latter case is hold, then C =D D C is supersolvable, so G=L D CL=L Š C =C \ L D C is supersolvable.
Assume that jLj < jU j. If p D q, then L < H , where jH j D jU j. By the hypothesis, there is a normal Â -pair .C; D/ for H and C =D is supersolvable. Clearly, L Ä D Ä C . Now Lemma 2.1 implies that .A; B/ WD .C ; D/ is a normal Â -pair for H and A=B Š C =D is supersolvable. If p ¤ q. Let Q =L be a Sylow subgroup of N=L, then by Schur-Zassenhaus theorem, it is esay to prove that Q D L Ì Q, where Q 2 Syl q .N /. By the hypothesis, Q has a subgroup U such that 1 < jU j < jQj and every subgroup H of Q of order jU j has a normal Â -pair .C; D/ such that C =D is supersolvable. Set B WD .CL \ HL/ G and observe that H D HL < G, L -D and BH is a proper subgroup of G. Now Lemma 2.2 implies the existence of a normal Â -pair .A; B/ for H with A=B supersolvable. Using Lemma 2.1, we find that .A; B/ is a Â -pair for H Ä G. Clearly, A=B Š A=B is supersolvable. In a word, we prove that there is a normal Â -pair .A; B/ of H with A=B supersolvable. Thus the hypothesis is hold for (G=L; N=L). By the minimality of jGj C jN j, the factor group G=L is supersolvable, as desired.
Step 3. Let q D max .N /, then N is q-closed.
Assume that N q is not normal in N and let N be a counterexample with least jGj C jN j for q-closed. As A=B is the homomorphic image of the subgroup A=H G of the supersolvable group C =H G , we find that .A; B/ is a normal Â -pair for H in N with A=B supersolvable. If N < G, then .N; N / satisfies the hypothesis, thus N is supersolvable by the minimality of jGj C jN j, so N is q-closed. If N D G, we can assume that fG r j r 2 .G/g is a Sylow system of G and K D G q G r for any r 2 .G/ with r 6 D q. By the similar discuss as above, the hypothesis is still true for .K; K/. If j .G/j 3, then N q C K, which implies that G q C G, a contradiction. Thus we may assume that jGj D jN j D p a q b .
Let L be a minimal normal subgroup of G, then G=L is q-closed by Step 2. Since q-closed is a saturated formation, we may assume that L -ˆ.G/ and L is the only minimal normal subgroup of G. If L is a q-group, then G q C G, a contradiction. Thus L Ä P and so L Ä O p .G/. We also get L is not cyclic by Step 2, so P is non-cyclic. Now we show that L D O p .G/. Let W be a maximal subgroup of G such that L 6 Ä W , then By the hypothesis, P has a subgroup U with jU j < jP j such that every subgroup H of order jU j has a normal Â -pair .C; D/ and C =D is supersolvable, then G D H C and D D .H \ C / G . If L Ä H , then G D H W , so W is a q-closed supplement of H in G. If L -H , then D D 1 by the minimal normality of L in G, so C is supersolvable, of course, is q-closed. Then every maximal subgroup of P has a q-closed supplement in G, so N D G is q-closed by Lemma 2.5, a contradiction.
Let q D max .N / and Q be a Sylow q-subgroup of N . Then by Step 3, Q is normal in N . By Step 2, we may assume that Q D N D P . Let L be a minimal normal subgroup of G contained in P . Then by the proof of Step 3, L D O p .G/ D P . But by Step 1, jLj Ä jU j < jP j, a contradiction. This contradiction completes the proof of this theorem.
Corollary 3.2 ([6, Theorem 3.1]). Assume that every maximal subgroup of any Sylow subgroups of a normal subgroup N of G has a normal Â -pair .C; D/ such that C =D is supersolvable, then G is supersolvable. Theorem 3.3. Assume that every non-cyclic Sylow subgroup P of G has a subgroup U with 1 < jU j < jP j such that every subgroup H of P of order jU j has a normal Â -pair .C; D/ with C =D supersolvable, then G is supersolvable.
Proof. The proof is similar to Theorem 3.1 and omitted here.
Corollary 3.4. Assume that every minimal subgroup of any non-cyclic Sylow subgroups of G has a normal Â-pair .C; D/ and C =D is supersolvable, then G is supersolvable.
Corollary 3.5. Assume that every maximal subgroup of any non-cyclic Sylow subgroups of G has a normal Â -pair .C; D/ and C =D is supersolvable, then G is supersolvable.
Theorem 3.6. Assume that every Sylow subgroup P of a normal subgroup N of G has a subgroup U with 1 < jU j < jP j such that every subgroup H of P of order jU j has a normal Â -pair .C; D/ with C =D nilpotent, then G is nilpotent.
Proof. Assume that the result is false and let G be a counterexample with least jGj. By Theorem 3.1, G is supersolvable and so is N . Let p D max .N /, P 2 Syl p .N /, then P C harN C G, so P C G.
Lemma 2.6 and the arguments in the Step 2 of the proof of Theorem 3.1 show that there is the unique minimal normal subgroup L of G contained in P such that G=L is nilpotent. By the hypothesis, P has a subgroup U with 1 < jU j < jP j such that every subgroup H of P of order jU j has a normal Â -pair .C; D/ and C =D is nilpotent. If H G D 1, then D D 1. Thus C is nilpotent, so is G D P C by C C G and P C G, a contradiction. Thus H G ¤ 1, so L Ä H G Ä H Ä P 1 , where P 1 É P , hence L Ä T P 1 ÉP P 1 Dˆ.P / Äˆ.G/. Then G=ˆ.G/ is nilpotent. Since the formation of nilpotent groups is saturated, we have G is nilpotent, a final contradiction.
Corollary 3.7 (see [6,Theorem 3.1]). Assume that every maximal subgroup of any Sylow subgroups of a normal subgroup N of G has a normal Â -pair .C; D/ such that C =D is nilpotent, then G is nilpotent.