# Groups whose primary subgroups are normal sensitive

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## Abstract

A subgroup \(H\) of a group \(G\) is said to be *normal sensitive* in \(G\) if for every normal subgroup \(N\) of \(H, N=H\cap N^{G}\). In this paper we study locally finite groups whose \(p\)-subgroups are normal sensitive. We show the connection between these groups and groups in which Sylow permutability is transitive.

## Keywords

Locally finite group Normal sensitivity Primary subgroup*PST*-group

*T*-group

## Mathematics Subject Classification (2000)

20E07 20E15 20F22 20F50## 1 Introduction

Let \(G\) be a group. A subgroup \(H\) of \(G\) is said to satisfy *the Congruence Extension Property* in \(G\) (or \(H\) is a *CEP-subgroup* of \(G\)) if whenever \(N\) is a normal subgroup of \(H\), there is a normal subgroup \(L\) of \(G\) such that \(N=H\cap L\). Equivalently, if for every normal subgroup \(N\) of \(H,\, N=H\cap N^{G}\). The definition of the *Congruence Extension Property* comes from category theory and has been studied on classical algebraic structures such as groups, rings, semigroups, monoids, etc (see for instance [8]). In Group Theory, this subgroup embedding property is called *normal sensitivity* and plays an important role in their structural study. In particular, if \(G\) is a finite group, every subgroup \(H\) of \(G\) with the character restriction property in \(G\), that is, every irreducible character of \(H\) is the restriction of some (irreducible) character of \(G\), is normal sensitive [7, Lemma 4(d)]. This fact, proved by Isaacs, is important to establish results about these type of subgroups as Berkovich showed in [7].

In connection with the above concept, Baumann [5] characterised the finite soluble \(T\)-groups as those groups in which every subgroup is normal sensitive. Here, as usual, by a \(T\)-*group* we mean a group in which normality is transitive, that is, \(H\unlhd \ K \unlhd \ G\) implies \(H\unlhd \ G\). Finite soluble \(T\)-groups were characterized by Gaschútz in 1957 (see [3, 2.1.11]). A group \(G\) is said to be a \(\overline{T}\)-*group* if every subgroup of \(G\) is a \(T\)-group. We remark that Baumann’s result has been extended to some classes of infinite groups by Bruno and Emaldi [9].

The motivation for the results of this paper comes from results of Beidleman and Ragland [6]. In that paper the authors established permutable and \(S\)-permutable versions of Baumann’s result. They indicated that normal sensitivity might shed light on the structure of locally finite groups. We recall that a subgroup \(H\) of \(G\) is said to be *permutable* (respectively, \(S\)-*permutable*) in \(G\) provided \(HK=KH\) for all subgroups (respectively, Sylow subgroups) \(K\) of \(G\). Similarly to \(T\)-groups, one defines \(PT\)-*groups* and *PST*-*groups* as those groups in which, respectively, permutability and \(S\)-permutability are transitive relations. The basic structures of finite soluble \(PT\)-groups and *PST*-groups have been established by Zacher and Agrawal, respectively (see [3, 2.1.11] and [3, 2.1.8]), while the structure of hyperfinite radical *PST*-groups and hyperfinite radical groups whose ascendant subgroups are \(S\)-permutable has been described by the authors in [4].

Let \(G\) be a radical hyperfinite *PST*-group and let \(L\) be the locally nilpotent residual of \(G\), that is the intersection of all normal subgroups \(N\) of \(G\) such that \(G/N\) is locally nilpotent. By [4, Theorem A], every \(p\)-subgroup of \(G\) is normal sensitive in \(G\) for all primes \(p\in \pi (L)\). Therefore it is natural to study the structure of groups whose \(p\)-subgroups are normal sensitive for different primes \(p\). The aim of the current paper is to study locally finite groups \(G\) whose \(p\)-subgroups are normal sensitive for every \(p\in \pi (L)\). Our main result is the following characterization.

### **Theorem A**

- (i)
\(2\not \in \pi (L)\);

- (ii)
\(L\) is abelian;

- (iii)
\(\pi (L)\cap \pi (G/L)=\emptyset \);

- (iv)
every subgroup of \(L\) is normal in \(G\); and

- (v)
\([H,G]=H\) for every non-identity subgroup \(H\) of \(L\).

Applying [12, Theorem PST] and Theorem A, we are able to obtain a description of the locally finite groups \(G\) whose \(p\)-subgroups are normal sensitive for every prime \(p\in \pi (L)\), in terms of \(S\)-permutability.

### **Corollary A1**

- (i)
Every \(p\)-subgroup of \(G\) is normal sensitive in \(G\) for every prime \(p\in \pi (L)\);

- (ii)
Every finite subgroup of \(G\) is a

*PST*-group; - (iii)
In every section of \(G\) the serial subgroups and the \(S\)-permutable subgroups coincide; and

- (iv)
Every section of \(G\) is a

*PST*-group.

### **Corollary A2**

Let \(G\) be a locally finite group and let \(L\) be the locally nilpotent residual of \(G\). If every \(p\)-subgroup of \(G\) is normal sensitive in \(G\) for every prime \(p\in \pi (L)\), then \(G\) is a radical hyperfinite *PST*-group.

### **Corollary A3**

Let \(G\) be a locally finite group and let \(L\) be the locally nilpotent residual of \(G\). If every \(p\)-subgroup of \(G\) is normal sensitive in \(G\) for every prime \(p\in \pi (L)\), then \(G\) is a radical hyperfinite group whose ascendant subgroups are \(S\)-permutable.

We are able to show that conditions of Corollaries A2 and A3 are also sufficient for a radical hyperfinite group to have its \(p\)-subgroups normal sensitive for every primes \(p\in \pi (L)\).

### **Corollary A4**

- (i)
\(G\) is a

*PST*-group; - (ii)
Every \(p\)-subgroup of \(G\) is normal sensitive in \(G\) for every prime \(p\in \pi (L)\);

- (iii)
Every ascendant subgroup of \(G\) is \(S\)-permutable in \(G\);

- (iv)
Every finite subgroup of \(G\) is a

*PST*-group; - (v)
In every section of \(G\) the serial subgroups and the \(S\)-permutable subgroups coincide; and

- (vi)
Every section of \(G\) is a

*PST*-group.

As Bruno and Emaldi [9] showed, locally finite groups whose subgroups are normal sensitive are locally finite \(\overline{{T}}\)-groups. We obtain that we can restrict normal sensitivity to primary subgroups of \(G\) (i.e. \(p\)-subgroups of \(G\), for some prime \(p\)) and still obtain \(\overline{{T}}\)-groups.

### **Corollary A5**

- (i)
Every primary subgroup of \(G\) is normal sensitive in \(G\);

- (ii)
Every subgroup of \(G\) is normal sensitive in \(G\); and

- (iii)
\(G\) is a \(\overline{{T}}\)-group.

## 2 Preliminary results

This section is auxiliar and is entirely devoted to give a proof of the different parts of our results. The first lemma reduces the normal sensitivity of subgroups of locally finite groups to finite subgroups.

### **Lemma 1**

Let \(H\) be a subgroup of a locally finite group \(G\). Then every subgroup of \(H\) is normal sensitive in \(G\) if and only if every finite subgroup of \(H\) is normal sensitive in \(G\).

### *Proof*

We recall that a group \(G\) is said to be *a Dedekind group* if every subgroup of \(G\) is normal. Baer [1] proved that every Dedekind group \(G\) either is abelian or has the form \(G = Q\times D\times B\), where \(Q\) is a copy of the quaternion group of order 8, \(D\) is an elementary abelian 2-subgroup and \(B\) is a periodic abelian subgroup whose elements have odd order. The next result shows that Sylow \(p\)-subgroups of locally finite groups are Dedekind groups provided every \(p\)-subgroup is normal sensitive.

### **Lemma 2**

Let \(G\) be a locally finite group and \(p\) be a prime. Suppose that every \(p\)-subgroup of \(G\) is normal sensitive in \(G\). If \(P\) is a Sylow \(p\)-subgroup of \(G\), then \(P\) is a Dedekind group. In particular, if \(p\) is odd, then \(P\) is abelian.

### *Proof*

We now show the normality of certain Sylow subgroups of the finite subgroups of the groups in consideration.

### **Lemma 3**

Let \(G\) be a locally finite group and \(p\) be a prime. Suppose that every \(p\)-subgroup of \(G\) is normal sensitive in \(G\). If \(K\) is a \(p\)-subgroup of \(G\) and \(L = N_G(K)\), then every subgroup of \(K\) is \(L\)-invariant.

### *Proof*

If \(H\le K\), by Lemma 2, every subgroup of \(K\) is normal in \(K\). Since \(K\) is normal sensitive in \(G\) and \(H\) is normal in \(K\), \(H = H^G\cap K\). On the other hand, \(H^L\cap K\le H^G\cap K\), so that \(H^L\cap K = H\). It follows that \(H\) is normal in \(L\). \(\square \)

### **Lemma 4**

Let \(F\) be a finite subgroup of a locally finite group \(G\). Let \(p\) be the least prime belonging to \(\pi (F)\). If every \(p\)-subgroup of \(G\) is normal sensitive in \(G\), then a Sylow \(p'\)-subgroup of \(F\) is normal in \(F\).

### *Proof*

*the*\({\mathfrak {X}}\)-

*residual*of the group \(G\). For example, if \(G\) is a group and \({\mathfrak {X}} = {\mathfrak {N}}\) is the class of all nilpotent groups, then the subgroup \(G_{\mathfrak {N}}\) of \(G\) while if \({\mathfrak {X}} = L{\mathfrak {N}}\) is the class of all locally nilpotent groups, then \(G_{L{\mathfrak {N}}}\) is the locally nilpotent residual of the group \(G\). We mention that \(G/G_{\mathfrak {N}}\) is nilpotent if \(G\) is finite and \(G/G_{L{\mathfrak {N}}}\) is locally nilpotent if \(G\) is locally finite. Furthermore, it is well know that if \(G\) is locally finite and \(\mathcal {L}\) is the family of all finite subgroups of \(G\), then we have

### **Lemma 5**

Let \(L\) be the locally nilpotent residual of a locally finite group \(G\). If every \(p\)-subgroup of \(G\) is normal sensitive in \(G\) for every \(p\in \pi (L)\), then \(G\) is locally supersoluble.

### *Proof*

Suppose now hat \(p_{m-1} = 2\). By Lemma 2, a Sylow \(2\)-subgroup of \(K\) is Dedekind. If \(D_{m-1}/D_m\) is abelian, we proceed in the same way and obtain that every \(K\)-chief factor of \(D_{m-1}/D_m\) has order \(2\). If \(D_{m-1}/D_m\) is not abelian, we have that \([D_{m-1}/D_m,D_{m-1}/D_m] = R_{m-1}/D_m\) has order \(2\) and the factor-group \(D_{m-1}/R_{m-1}\) is an elementary abelian \(2\)-group. Applying the previous arguments, we see that every \(K\)-chief factor of \(D_{m-1}/R_{m-1}\) has order \(2\). Hence \(D_{m-1}/D_m\) has a series of \(K\)-invariant subgroups whose factors have order \(2\).

Therefore in both cases, proceeding in the same way, after finitely many steps we obtain that \(D\) has a series of \(K\)-invariant subgroups whose factors have prime orders. Since \(K/D\) is nilpotent, \(K\) is supersoluble, as required. \(\square \)

Several features of our groups can now be deduced.

### **Corollary 1**

Let \(L\) be the locally nilpotent residual of a locally finite group \(G\). If every \(p\)-subgroup of \(G\) is normal sensitive in \(G\) for every \(p\in \pi (L)\), then \(2\notin \pi (L)\).

### *Proof*

By Lemma 5, \(G\) is locally supersoluble, that is every finite subgroup of \(G\) is supersoluble. From the known properties of finite supersoluble groups we obtain that, for each prime \(p\), the Sylow \(\Gamma _p\)-subgroup of \(G\) is normal, where \(\Gamma _p = \{q\in \pi (G)\ |\ q > p\}\). In particular, the Sylow \(2'\)-subgroup \(H\) of \(G\) is normal. Since \(G/H\) is a locally finite \(2\)-group, \(G/H\) is locally nilpotent. It follows that \(L\le H\). In particular, \(L\) is a \(2'\)-subgroup. \(\square \)

### **Corollary 2**

Let \(L\) be the locally nilpotent residual of a locally finite group \(G\). If every \(p\)-subgroup of \(G\) is normal sensitive in \(G\) for every \(p\in \pi (L)\), then \(L\) is locally nilpotent.

### *Proof*

Let \(K\) be an arbitrary finite subgroup of \(G\). By Lemma 5, \(G\) is locally supersoluble and so \(K\) is supersoluble. From the known properties of finite supersoluble groups we obtain that \([K,K]\) is nilpotent. The obvious inclusion \(K_\mathfrak {N}\le [K,K]\) gives that \(K_\mathfrak {N}\) is nilpotent. If \(\mathcal {L}\) is the family of all finite subgroups of \(G\), we have already remarked that \(L = \cup _{K\in \mathcal {L}}K_\mathfrak {N}\). If follows that \(L\) is locally nilpotent. \(\square \)

### **Corollary 3**

Let \(L\) the locally nilpotent residual of a locally finite group \(G\). If every \(p\)-subgroup of \(G\) is normal sensitive in \(G\) for every \(p\in \pi (L)\), then \(L\) is abelian and every subgroup of \(L\) is \(G\)-invariant.

### *Proof*

### **Corollary 4**

Let \(L\) be the locally nilpotent residual of a locally finite group \(G\). If every \(p\)-subgroup of \(G\) is normal sensitive in \(G\) for every \(p\in \pi (L)\), then \([H,G] = H\) for every non-identity subgroup \(H\) of \(L\).

### *Proof*

The last feature we consider needs more effort.

### **Lemma 6**

Let \(L\) be the locally nilpotent residual of a locally finite group \(G\). If every \(p\)-subgroup of \(G\) is normal sensitive in \(G\) for \(p\in \pi (L)\), then \(\pi (L)\cap \pi (G/L) = \emptyset \).

### *Proof*

## 3 Proof of the main results

We are in a position to deduce easily the results of this paper.

### *Proof of Theorem A*

Necessity of the conditions. Assertion (i) has been proved in Corollary 1, assertions (ii) and (iv) in Corollary 3, assertion (iii) in Lemma 6 and assertion (v) in Corollary 4.

Conversely, suppose that \(G\) is a locally finite group that satisfies conditions (i)–(v). Let \(p\) be a prime and let \(P\) be a \(p\)-subgroup of \(G\). Suppose that \(p\in \pi (L)\). By (ii), \(P\) is abelian. Let \(K\) be a subgroup of \(P\). Condition (iv) shows that \(K\) is normal in \(G\), that is \(K = K^G\) and hence \(P\cap K^G = P\cap K = K\). \(\square \)

### *Proof of Corollary A2*

If every \(p\)-subgroup of \(G\) is normal sensitive in \(G\) for every \(p\in \pi (L)\), by Theorem A, \(G\) satisfies the conditions (i)–(v) of the statement of this result. By [12, Theorem PST], \(G\) is a *PST*-group. \(\square \)

### *Proof of Corollary A3*

As above application of [12, Theorem PST] gives that every serial subgroup of \(G\) is \(S\)-permutable in \(G\). In particular, every ascendant subgroup of \(G\) is \(S\)-permutable in \(G\).\(\square \)

### *Proof of Corollary A4*

If \(G\) is a *PST*-group, by [4, Theorem 4.7], \(G\) satisfies the conditions (i)–(v) of Theorem A. Therefore every \(p\)-subgroup of \(G\) is normal sensitive in \(G\) for every \(p\in \pi (L)\), that is (i) implies (ii). By Corollary A3, (ii) implies (iii). Finally, suppose that every ascendant subgroup of \(G\) is \(S\)-permutable in \(G\). Let \(U\) and \(V\) subgroups of \(G\) such that \(U\) is an \(S\)-permutable subgroup of \(V\) and \(V\) is an \(S\)-permutable subgroup of \(G\). By [4, Proposition 2.4], \(V\) is an ascendant subgroup of \(G\) and \(U\) is an ascendant subgroup of \(V\). Hence \(U\) is an ascendant subgroup of \(G\), and it follows that \(G\) is a *PST*-group. That is, (iii) implies (i). The remainder follows from Corollary A1. \(\square \)

### *Proof of Corollary A5*

## Notes

### Acknowledgments

This research was supported by Proyecto MTM2010-19938-C03-01 (Ballester-Bolinches, Pedraza) and Proyecto MTM2010-19938-C03-03 (Kurdachenko, Otal) from MINECO (Spain). The third author was also supported by Gobierno of Aragón (Spain) and FEDER funds.

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