SOME RESULTS ON-SOLVABLE AND SUPERSOLVABLE GROUPS

For a finite group G, Cp(G), Sp(G), L(G) and S(G) are generalizations of the Frattini subgroup of G. We obtain sne results on #-solvable, l>-solvable and supersolvable groups with the help of the structures of these subgroups. KEY I3RDS AND PHRASES. p-solvable, IT-solvable, supersolvable. 1991 /S SUBJEL’T CLASSIFICATION COEES. Primacy 20[]10, 20EIZ5; Secondary 20F160 20E20. 1. INTROEUCTION. Many authors have considered various generalizations of the Frattini subgroup of a finite group. Deskins [6] considered the subgroup p(G), Mukherjee and lattacharya [4] the subgroup Sp(G) and hatia [3] the subgroup L(G). In [?], we introduced the subgroup S(G) and investigated its influence on solvable group. In this paper, our aim is to prove scxne results which imply a finite group G to beTT’-solvable, p-solvable and supersolvable. All groups are assumed to be finite. We use standard notations as found in Gorenstein [8] and denote a maximal subgroup M of G b M G. 2. PRELIMINARIES. EEFINITION. Let H and K be two normal subgroups of a group G with KocH. Then the factor group H/K is called a chief factor of G if there is no n6rmal subgroup N of G such that KCNH, with proper inclusion. Let M be a maximal subgroup of G. Then H is said to be a normal supplement of M in G if t4t G. The normal index of M in G is defined as the order of a chief factor H/K, where H is minimal in the set of all normal supplements of M in G and is denoted b G M). (2.1) (Deskins [6,(2.1)], Beidleman and Spencer [2, Lemm-1]) If M is a maximal subgroup of a group G then I(G M) is uniquely detemined. (2.2) (Beidleman and Spencer [2, Lemm-2]) If N is a normal subgroup of a group G and M is a maximal subgroup of G such that NC_ M then {G/N M/N) .(G M) (2.3) (MukherJee [9, Theorem-I]) If M is a maximal subgroup of a group G and MnG then (G:M)=[G:M]=a prim. (2.4) (Baer [1, Lemna-3]) If the group G possesses a maximal subgroup with core I then the following properties of G are equivalent. 60 T.K. DUTTA AND A. BHATTACHARYYA (I) The indices in G of all the maximal subgroups with core 1 are powers of one and the same prime p. (2) There exists one and only one minimal normal subgroup of G and there exists a crmon prime divisor of all the indices in G of all the maximal subgroups with core I. (3) There exists a non-trivial solvable normal subgroup of G. _2INITION.’Let’G be a group and p beany prime. The four characteristic subgroups of G, which are analogous to the Frattini subgroup {G), are defined as follows


INTROEUCTION.
Many authors have considered various generalizations of the Frattini subgroup of a finite group.Deskins [6] considered the subgroup p(G), Mukherjee and lattacharya  [4] the subgroup Sp(G) and hatia [3] the subgroup L(G).In [?], we introduced the subgroup S(G) and investigated its influence on solvable group.In this paper, our aim is to prove scxne results which imply a finite group G to beTT'-solvable, p-solvable and supersolvable.All groups are assumed to be finite.We use standard notations as found in Gorenstein [8] and denote a maximal subgroup M of G b M G. 2.
PRELIMINARIES.EEFINITION.Let H and K be two normal subgroups of a group G with KocH.Then the factor group H/K is called a chief factor of G if there is no n6rmal subgroup N of G such that K CNH, with proper inclusion.Let M be a maximal subgroup of G. Then H is said to be a normal supplement of M in G if t4t G.The normal index of M in G is defined as the order of a chief factor H/K, where H is minimal in the set of all normal supplements of M in G and is denoted b G M).
(2.2) (Beidleman and Spencer [2, Lemm-2]) If N is a normal subgroup of a group G and M is a maximal subgroup of G such that NC_ M then {G/N M/N) .(GM) (2.3) (MukherJee [9, Theorem-I])   If M is a maximal subgroup of a group G and MnG then (G:M)=[G:M]=a prim.
(2.4) (Baer [1, Lemna-3])   If the group G possesses a maximal subgroup with core I then the following properties of G are equivalent.
(I) The indices in G of all the maximal subgroups with core 1 are powers of one and the same prime p.
(2) There exists one and only one minimal normal subgroup of G and there exists a crmon prime divisor of all the indices in G of all the maximal subgroups with core I.
(3) There exists a non-trivial solvable normal subgroup of G. _2INITION.'Let'Gbe a group and p beany prime.The four characteristic subgroups of G, which are analogous to the Frattini subgroup {G), are defined as follows where In case G) is empty then we define G (G) and the same thing is done for the other three subgroups.
(2.5) If H is a subgroup with finite index n in a group G then coreGH has finite index dividing n! (2.6) (Butta and lhattacharyya [7, Theorem-3.5]) If G is p-solvable then SjtG) is solvable.
EEFINITION.Let M be a maximal subgroup of a group G. Then M is said to be c- maximal if [G:M] is ccmposlte.3.
S( RESULTS ON p-SOLVABLE AND If-SOLVABLE GROUPS. 3.1.Let p be the largest prime dividing IGI anti p(G) # .Then G is p- PIKI3F.Let G satisfy the hypothesis of the theorem.Then G is not simple.For, otherwise IGlp I(G'M)p [G:M]p I, where M belongs to ,-n(G)' which contradicts the fact that p divides IGI.Let N be a minimal normal subgroup of G.If p does not divide IG/NI then G/N is a p'-group and hence it is p-solvable.If p divides IG/NI then p is the largest prime dividing IG/NI.If 'p(G/N) then G/N Sp(G/N).B Theorem-8(i) [I0], Sp(G/N) is solvable and hence G/N is p-solvable.now assume that p(G/N) # .
By Uemm-2 [2], we obtain (G/N:M/N)p [G/N M/NIp for each M/N in p(G/N).So by induction, G/N is p-solvable.We note that Sp(G) # G, since ,-'(G) # .If NqSp(G) then N is solvable and so it is p-solvable,and consequently G is p-solvable.If  I and so N is p-solvable and hence G is p-solvable.P The converse follows directly from Theorem I [2].THEDREM 3.2.Let p be the largest prime dividing IGI.Then G is p-solvable if the following hold."rG'M 1)p "I(G'M2) p then [G'M 1]p [G'M2] p REMARK 3.3.The converse of the above theorem is not necessarily true.Let G be a p-group, where p is any prime.Then G is p-solvable, but it has no c-maximal subgroup and so G does not satisfy the hypothesis (i) of the above theorem.If the group G has a c-maximal subgroup then the converse of Theorem 3.2 follows from Theorem 1 [2].THEOREM 3.4.Let G be a p-solvable group and G) # .Then G is Let the condition of the theorem hold.Let G be simple.Then it inmediately follows that either G is a p'-group or is of prime order p.If G is of prime order p then it is solvable and hence[-solvable.If G is a p'-group then IGlp I. Also IGI is composite.For, otherwise, G is cyclic and hence it is-solvable.Let IGlrr# 1 and Pl' P2 Pn be the set of prime divisors of IGI, which belong to 11".Let S(Pi) (i 1,2 n) denote the Sylow Pi-subgroup of G. Then S(Pi) # G for 1,2 n.For, otherwise, G is solvable and hence G is/T-solvable.Let M be the maximal subgroups of G such that S(Pi).CMiCG and so [G:M i]p.
As each pi G rl, it follows that IGlrr l, a contradiction.So IGlt=l and hence G is l/-solvable.I/e now suppose that G is not simple.
Let N be a minimal normal subgroup of G. Then G/N is a p-solvable group.If .(G/N)then G/N (G/N) and so by (2.6), it follows that G/N is solvable and hence it is if-solvable.We now assume that [(G/N) # .Using Lemma 2 [2], we obtain (G/N M/N)r[ [G/N M/N]ff for each M/N in _99(G/N).By induction, G/N is E'-solvable.Let N 1 be another minimal normal subgroup of G. Then G/N 1 is iT-solvable.Since G G/N t%N 1 is isomorphic to a subgroup of the /r-solvable group G/N x G/N 1, it follows that G is solvable.We may now assume that N is the unique minimal normal subgroup of G.We shall now show that N is l-solvable.We note that SG) # G, since (G) # .If N( S(G) then by (2.6) it follows that N is solvable and hence it is if-solvable.If NeSt(G) then there exists M in X(G) such that N M and so G MoN and co.reG(Mo) 41>.Let M be any maximal subgroup of G with core 1.Then NM and so G .Clearly M belongs to

,(G). By hypothesis INlrl
If INII # I then there exists a comnon prime divisor of all the indices in G of all the maximal subgroups with core I.So by (2.4), N is solvable and hence it is -solvable.
Thus G/N and N are both 1J-solvable.So G is W-solvable.The converse follows directly from Theorem 2 [9].
PROOF.Let the condition of the theorem hold.If IGIw I then G is a '-group and hence it is g-solvable.So we assume that IGly # 1.Let G be simple and Pl' P2 Pn be the set of prime divisors of IGI, which belong to ff.Then as in the proof of Theorem 3.4, we can show that there exist maximal subgroups M of G such that [G:M i]

By hypothesis, IGlff [G:MI] = [G.M2}
[G:Mn]ff.As each pi e, it follows that IGlff 1, a contradiction.So G can not be simple.Let N be a minimal normal subgroup of G. If(G/N) is empty then A(G/N) is also empty and so Oy definition, L(G/N) G/N and consequently by the supersolvability of the group L(G/N), it follows that G/N is- solvable.If (G/N) consists of only one element M/N, say, then either A(G/N) is empty orA(G/N) IM/NI.If A(G/N) is empty then as above G/N is supersolvable.If (G/N) M/N then M/N L(G/N) and consequently M/N is normal in G/N.So by Theorem 1 [9], a prime, a contradiction, since M/NA(G/N).We now assume that I(G/N)I 2. It can be shown that G/N satisfies the hypothesis of the theorem.So by induction, G/N is if-solvable.As before, we can assume that N is the unique minimal normal subgroup of G. Also we see that L(G) # G.If NL(G) then N is solvable and hence it is -solvable.If NL(G) then there existsM in A(G) such that NM and so G MoN and coreG(Mo) I. Let M be any maximal subgroup of G with core 1.Then NM and so G t,q. Consequently (G:M)=INl=[G:Mo),whence it follows that M belongs to (G  TH]K)REM 3.9.Let G be a group with I(G)I 2. Then G is -solvable if and only if the following hold.PROPOSITION 3.12.Let G be a p-solvable group or p be the largest prime dividing and[p(G) # .Then G is if-solvable if'.(G'M)nI for each M in p(G).
PROPOSITION 3.13.Let G be a group with I%(G)I 2. Then G is W-solvable if (G:M1) ff "t[G'M2) ff 1 for all M 1, M 2 belonging to (G) with equal normal index.PROOF.Let G satisfy the hypothesis of the proposition.Then G is not simple.
For, otherwise, [G]ff L[G'M)ff I and so G is if-solvable.Let N be a minimal normal subgroup of G.If N%M then N isCr-solvable and also, by induction, G/N is /'-solvable and hence G is r/'-solvable.If NM then G=tvq and since G/N M/MaN, G/N is ll-solvable.Also by hypothesis INlr/ (G'M)7 I and so N is [-solvable.Hence G is W-solvable.

4.
SCME RESULTS ON SUPERSOLVABLE GROUPS.THEDREM 4.1.Let G be a p-solvable group and suppose that for each c-maximal a p-solvable c-maximal subgroup M with *G'M) [G'M] P P If M 1 and M 2 are c-maximal subgroups of G with a Y'-solvable maximal subgroup M with (G'M)=[G'M].(ii)(G'MI)=(G:M2) implies [G'MI]=[G'M2] for any MI,M 2 in )(G).

PROFITION 3. 11 .
Let G be a group with/(G) # i.Then G. is 17-solvable if .(G'M)rrI for each M in A(G).

PROPOSITION 3 .
14.If a group G has a N-solvablemaximal subroup M with (G:M)ff then G is if-solvable.
).By hypothesis [G'M]n=INI.If IN[n=1 then N is if-solvable.If INI then using (2.4), we have N is solvable and hence it is if-solvable.Thus G/N and N are both if-solvable and consequently G is W-solvable.The converse follows directly frgn Theorem 5 [9].