A Note on Finite Groups in which C-Normality is a Transitive Relation

A subgroup H of G is c−normal in G if there exists a normal subgroup N of G such that HN = G and H ∩N ≤ HG. A group G is called


INTRODUCTION
All groups considered in this paper are finite.Notations and terminology not explained can be found in Doerk and Hawkes [4], Robinson [1] and Wang [7].This paper will focus mainly on characterizing several classes of solvable groups.More specifically, we will consider the class of solvable groups in which c−normality is a transitive relation.First a few definitions and some known results should be discussed.
A subgroup H of a group G is said to permute with a subgroup K if HK is a subgroup of G. H is said to be permutable in G if it permutes with all the subgroups of G.We write H per G to denote H is permutable in G.Among the first studies on permutable subgroups was done by Ore in 1939.Ore used the term qusinormal subgroups to represent permutable subgroups.Kegel showed that such subgroups are necessarily subnormal.Actually, a result stronger than permutable subgroups are subnormal subgroups.For a subgroup H of G, it is enough to know that H permutes with all of its own conjugate to deduce that H is subnormal [2].On the other hand subnormality does not imply permutability.As an example, one might consider the alternating group of order 12.It has a subnormal subgroup of order 2 that is not permutable.A subgroup H is called S−permutable in G.This concept was introduced by Kegel in 1962 [3], who showed such subgroups are necessarily subnormal.
A group G is said to be a It is a direct consequence of the subnormality of permutable (S−permutable) subgroups that P T −groups (P ST -groups) are precisely those groups in which subnormality and permutability (S−permutability) coincide.
Let us consider the following definitions.

Definition 1 [7]
Let H be a subgroup of a finite group G.The core of H in G , denoted as H G ,is defined to be the largest normal subgroup of G contained in H, (or equivalently to H G = ∩{H g : g ∈ G}).
Definition 2 [7] Let G be a group.We will call a subgroup While it is clear that normal subgroups are c−normal, the converse is not true.For example the Sylow 2−subgroup H of the symmetric group S 3 is c-normal in S 3 but H is not even subnormal in S 3 .
Stimulated by the theory of T −groups (resp.P T −groups, P ST −group), we introduce the class of CT −groups..
3. A solvable CT −group need not be a T −group and a T −group need not be a CT −group.
The following example shows that a solvable CT −group need not be a T −group.

PRELIMINARIES
In the years 1953,1964, and 1975, Gaschutz, Zacher, and Agrawal respectively, proved the following definitive results on solvable T −groups, P T −groups, and P ST −groups.
Theorem 2 [3] [6] Let G be a group with L the nilpotent residual of G. Then G is a solvable P T −group ( resp.P ST −group), if and only if the following condition hold: (i) L is normal abelian Hall subgroup of G with odd order; (ii) G/L is Dedekind ( nilpotent ) group; (iii) G acts by conjugation as power automorphism on L.
The following known results about c−normal subgroups will be used in this paper several times.
Lemma 3 [7] Let G be a group.Then Lemma 4 [7] Let G be a finite group.Then G is solvable if and only if every maximal subgroup of G is c−normal in G.
The following Lemma is a direct of consequence of Lemma 4 and definition of CT −groups.

Lemma 5 Every Maximal subgroup of a solvable
Proof.Let N be a minimal normal solvable subgroup of G, then |N | = p n for some prime p. Hence every subgroup of N is c−normal in G. Let P be a Sylow p−subgroup of G. Then N is a normal subgroup of P and N ∩ Z(P ) = 1.Let x be a nonidentity element N ∩ Z(P ).Assume that Lemma 7 Let G be a solvable CT −group.Then: (ii) follows from Lemma 3

MAIN RESULTS
The following theorems state our main results, and show evidence that CT −groups are quite close to T −groups, P T −groups, and P ST −groups.
is a CT −group.Proof.For (i) let H be subgroup of a CT −group G.We consider two cases for H: Case 1.If H is maximal in G then by Lemma 5, H is a CT −group.Case 2. Suppose H is not maximal in G. Let M be a maximal subgroup of G containing H. By Lemma 5 M is a CT −group.By induction, every subgroup of M is a CT −group .Hence H is a CT −group.
To prove the theorem we must show thatH is c−normal in G. Since K is c−normal in G then there exists a normal subgroup N of G such that G = KN and K ∩ N ≤ K G .From (|G 1 |, |G 2 |) = 1 we get N N 1 × N 2 where N i = (G i ∩ N) and K G = K 1G × K 2G with K iG = (K G ∩ N) for each i ∈ {1, 2}.It is clear now that c−normality of K in G is equivalent to the c−normality of K i in G i .Arguing in a similar way, we get H H 1 × H 2 with H i c−normal in K i for each i ∈ {1, 2}.Therefore H is c−normal in G.
Theorem 8 A solvable CT −group is supersolvable.Proof.Suppose that the statement is false and let G be a counter example of minimal order.Let N be a minimal normal subgroup of G.By Lemma 6, N is cyclic.By part (ii) of Lemma 7, G/N is a CT −group.Therefore G/N is a supersolvable group by induction.Hence G is supersolvable.Theorem 9 If G 1 and G 2 are two CT −groups and (|G 1 |, |G 2 |) = 1, then G = G 1 × G 2 is also a CT −group Proof.Let H and K be subgroups of G such that H is c−normal in K and K is c−normal in G.Theorem 10 Let G be a group with L the nilpotent residual of G.If G is a solvable CT −group, then the following conditions hold: