A NOTE ON pSOLVABLE AND SOLVABLE FINITE GROUPS 821

The notion of normal index is utilized in proving necessary and sufficient conditions for a group G to be respectively, p-solvable and solvable where p is the largest prime divisor of |G|. These are used further in identifying the largest normal p-solvable and normal solvable subgroups, respectively, of G.


INTRODUCTION AND NOTATION.
Structures of solvable and p-solvable finite groups are closely related to the indices and normal indices of various kinds of maximal subgroups.The largest and also the smallest prime divisors of the order of a group seem to play, in this connection, important roles in the investigation of these structures.This is precisely the focus of the present note.
The following standard notations and terminologies have been used throughout.
(a) M is a maximal subgroup of G: M < .G (b) Normal index of a maximal subgroup M: r/(G:M) (c) The p-part of normal index: r/(G: M)p (d) The p-part of index of maximal subgroup M of G: [G: M], (e) p(G) V {M < .a[a: M]p 1} (f) A maximal subgroup whose index is a composite number: c-maximal subgroup 2. PRELIMINARIES.
If % is a minimal normal supplement to M < .G then for any chief factor %/K, K C M and G M%. Evidently, [G:M] divides I%/gl= r/ G M and if G is simple then obviously r/(G:M) G], VM < .G. the integer r/(G: M)is unique VM < .G. [2].
LEMMA 2.2 [5, Theorem 3]   In any group G the following are equivalent.
(3) r/(G:M) is power of a prime for all Inaximal subgroups M of G.
(4) r/(G: .)=[G: M] for all maximal subgroups M of G. LEMMA 2.3.[1, Lemma 3]   If G is a group with a maximal core free subgroup then the following are equivalent: (i) There exists a unique minimal normal subgroup of G and there exists a common prime divisor of the indices of all maximal core free subgroup of G.
(ii) There exists a nontrivial solvable normal subgroup of G.
(ii) The indices of all maximal core free subgroup of G are powers of a unique prime.

THEOREM 2.4 [6, Theorem 8]
If p is the largest prime dividing the order of a group G then Sr,(G fq {M < .GIM is c- maximal and [G: M], 1} is solvable.
While the equality of the indices and normal indices of each maximal subgroup M of G is both necessary and sufficient for G to be solvable, rI(G:M),= [G:M], 'v'M <.G does not necessarily imply G is p-solvable.This condition holds in G PSL(2, 7) for p 2. However G is not 2-solvable.
THEOREM 3.1.Let p be the largest prime divisor of the order of a group G. Then G is p- solvable if and only if r/(G: M), [G: M], V c-maximal subgroup M of G.
PROOF.Let M be a c-maximal subgroup of the p-solvable group G and consider GIN where N is a minimal normal subgroup of G. Case NI pr.If N C M then by induction it follows that rl(G/Y: M/Y)r, [G/N: M/N], i.e., r/(G: M), [G: M],.On the other hand if N:M then q(G:M)p=[G:M],=I since G=MN.Case II.NI =p'.Observe that N is elementary abelian and if N C M then r/(G:M), IN [G:M],.Now, suppose N C M and consider GIN.If p divides G/N then the equality r/(G:M)p [G:ML, follows by induction and if p [GIN[then -is a p'-group and trivially, rI(G[N:M/N),=[G/N:M[N],, i.e., r/(G: M), [G: M],.
Conversely, let rl(G: M) [G: M]pV c-maximal subgroup of Go Step I. G is not simple.Let K < .G and [G: Kip 1. K cannot be c-maximal as otherwise G is simple implies GI, and trivially G is p-solvable.Suppose [G:K] q, a prime.
By representing G on the cosets of K it follows that core K 1 and so G cannot be simple.Let N be a minimal normal subgroup of G and consider GIN.
II. " is p-solvable.If p divides G/N[ and is a c-maximal subgroup of GIN then Step rl(G/Y: M/N)r, [G/N: M/NIt and by induction GIN is p-solvable.Observe that, if GIN has no c-maximal subgroup then GIN is supersolvable and so GIN is p-solvable.However, if p] G/N then GIN is a p'-group and trivially, therefore GIN is p-solvable.
Step III.G is p-solvable.If N C S,(G) then by theorem 2.4 [6] it follows that N is solvable and therefore is elementary abelian.Consequently, G is p-solvable.If N S,(G) then G MN, [G: M], and M is a c-maximal subgroup of G.This implies r/(G: M)p Yl [G: M]p 1, i.e., g is a p'-group and G is consequently, p-solvable.Theorem 3.1 can be used to identify the largest normal p-solvable subgroup of G when p is the largest prime divisor of G].PROOF.If G has no c-maximal subgroup then G is supersolvable and on the other hand if for each c-maximal subgroup M of G. r/(G:M), [G:M], then by theorem 3.1 G is p-solvable.
Therefore, for C 0, the assertion in the theorem follows since in that event T G. Now, suppose C and N is a minimal normal subgroup of G included in T. If p does not divide G/N[ then G/N is trivially p-solvable and so T/N is p-solvable.Otherwise, by induction, it follows that TIN is p-solvable and N may be treated as the only minimal normal subgroup of G in T. If V N is another minimal normal subgroup of G then consider ='G Suppose, p divides 17 I, and set C*={--= is c-max in IrI(G'X),#[G'X]p}.If has no c- maximal subgroup then is supersolvable.This implies -T is supersolvable and consequently, T is p-solvable.Again, if V c-max subgroup R in , r/(G" X)p [G" X]p then by theorem 3.2, =is p-solvable and this will imply, as before, T is p-solvable.C* may therefore be assumed nonempty.Set -= f {X is c-max in I" E C*}.Note that if X C* then X C and this implies R D T. By induction R/V is p-solvable and therefore TV _ T T is p-solvable.
V -TV- [G/V then G/V is trivially p-solvable and consequently, T is p-solvable, N may therefore be viewed as the unique minimal normal subgroup of G.It may be assumed that p divides IN[ as otherwise N is a p-group and p-solvability of T follows.
Since is solvable ([4], 1.1) one may assume i C ep and G= YN, Y < .G,[G:Y],= 1.If [G:Y] is composite then r/(G:Y), NI, [G:Y], 1 and N is a pr-group.This implies, however, that T is p-solvable.Assume, therefore, [G:Y] q, a prime.By representing G on the cosets of Y it follows that core Y # 1 and this contradicts the fact that the unique minimal normal subgroup N C Y. Consequently, it now follows that T is p-solvable.
We shall now show that T is indeed the largest normal p-solvable subgroup of G. Suppose K is the largest normal p-solvable subgroup of G and let D be a minimal normal subgroup of G in K. Then D is either a pr-group or is an elementary abelian p-group.If D is a p-group and M C then D C M implies G MD.But then r/(G: M), [G: M], 1, a contradiction and so D is included in each M C. Similarly, if D is an elementary abelian p-group then also VM C,D C M. By induction, =--, i.e., T K and the assertion in the theorem is proved completely.(If p I-l then trivially G is p-solvable and T G).
Applying similar techniques it is not difficult to prove the following results.THEOREM 3.3.Let p be the largest prime divisor of the order of a group G and C be the class {M < .GI[G:M], 1} of maximal subgroups of G. Then G is p-solvable if and only if r/(G: M), I'M C.
COROLLARY.V {M C lrl(G: M), 1} S is p-solvable.By induction it can be shown as in above that S is indeed the largest normal p-solvable subgroup of G. 4. SOLVABILITY CONDITIONS.
If the indices of all maximal subgroups of a group G is prime then it is well known that G is supersolvable.G turns out to be solvable if only a subclass of maximal subgroups has prime indices.THEOREM 4.1.Let C be the class {M < .GI[G:M], 1, p is the largest prime divisor of G[} of maximal subgroups of a group G. Then G is solvable if each maximal subgroup in C has prime index.

PROOF. Note, C
and if M E C then by representing G on the cosets of M it follows that G is not simple.Consider GIN where N is a minimal normal subgroup of G.If pXIG/NI then v--M N < .G/N,[G/N:M/N]r,= 1, i.e., [G:M]p which however implied [G/N:M/N] is of prime index.Consequently, GIN is supersolvable.On the other hand, if p divides G/NI then by induction GIN is solvable.
N may therefore be treated as the unique minimal normal subgroup of G.If N 45, then G MN and by representing G and the cosets of M it follows that core M 1 and we have a contradiction.Hence N C , and therefore G is solvable.
REMARK.The theorem does not hold if p is note the largest prime divisor.In PSL(2, 7)= G,'v'M < .G and [G: M] 1, [G: M] a prime.But G is not solvable.
The equality of the index and the normal index VM < .G is both necessary and sufficient for the solvability of a group G.The result remains valid if this holds for the subclass of maximal subgroups with odd indices.The above theorem can be easily proved using induction and the fact that every odd ordered group is solvable.
In proving the corollary, one uses the same techniques as in the proof of theorem 3.2.That W is the largest normal solvable subgroup of G follows from lemma 3 in [10].For the sake of completeness Lemma 3 mentioned above is stated below.LEMMA.In any group G, W is the largest normal solvable subgroup.REMARKS.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable:

THEOREM 3 . 2 .
Let C denote the class {M is c-maximal in a lr(G.M)p # [G: M],, p is the largest prime divisor of [GI of maximal subgroups of a group G. Then T N {M < .G[M E C} is the largest normal p-solvable subgroup of G.