Monatshefte für Mathematik

, Volume 175, Issue 2, pp 175–185

# Groups whose primary subgroups are normal sensitive

• Leonid A. Kurdachenko
• Javier Otal
• Tatiana Pedraza
Article

## 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

Let $$G$$ be a locally finite group and let $$L$$ be the locally nilpotent residual of $$G$$. Then every $$p$$-subgroup of $$G$$ is normal sensitive in $$G$$, for every prime $$p\in \pi (L)$$, if and only if $$G$$ satisfies the following conditions:
1. (i)

$$2\not \in \pi (L)$$;

2. (ii)

$$L$$ is abelian;

3. (iii)

$$\pi (L)\cap \pi (G/L)=\emptyset$$;

4. (iv)

every subgroup of $$L$$ is normal in $$G$$; and

5. (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

Let $$G$$ be a locally finite group and let $$L$$ be the locally nilpotent residual of $$G$$. Then the following statements are equivalent:
1. (i)

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

2. (ii)

Every finite subgroup of $$G$$ is a PST-group;

3. (iii)

In every section of $$G$$ the serial subgroups and the $$S$$-permutable subgroups coincide; and

4. (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

Let $$G$$ be a radical hyperfinite group and let $$L$$ be the locally nilpotent residual of $$G$$. Then the following statements are equivalent:
1. (i)

$$G$$ is a PST-group;

2. (ii)

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

3. (iii)

Every ascendant subgroup of $$G$$ is $$S$$-permutable in $$G$$;

4. (iv)

Every finite subgroup of $$G$$ is a PST-group;

5. (v)

In every section of $$G$$ the serial subgroups and the $$S$$-permutable subgroups coincide; and

6. (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

Let $$G$$ be a locally finite group. Then the following statements are equivalent:
1. (i)

Every primary subgroup of $$G$$ is normal sensitive in $$G$$;

2. (ii)

Every subgroup of $$G$$ is normal sensitive in $$G$$; and

3. (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

The necessity is trivial. Suppose that every finite subgroup of $$H$$ is normal sensitive in $$G$$. Let $$K$$ be an arbitrary subgroup of $$H$$ and $$T$$ be a normal subgroup of $$K$$. Let $$\mathcal {L}$$ be the family of all finite subgroups of $$K$$. We have
\begin{aligned} T^G\cap K = T^G\cap (\cup _{F\in \mathcal {L}}F) = \cup _{F\in \mathcal {L}}(T^G\cap F). \end{aligned}
Clearly $$T\cap F$$ is normal in $$F$$. Since $$F$$ is normal sensitive in $$G,\, T\cap F = (T\cap F)^G\cap F$$. Pick $$x\in \cup _{F\in \mathcal {L}}(T^G\cap F)$$. Then there exists a finite subgroup $$F_0$$ of $$K$$ such that $$x\in T^G\cap F_0$$. Since $$x\in T^G$$, there exist $$y_1,\ldots , y_m\in T$$ and $$g_1,\ldots , g_m\in G$$ such that $$x = y_1^{g_1}\cdots y_m^{g_m}$$. Put $$F_1 = \langle F_0, y_1,\ldots , y_m\rangle$$ so that $$F_1$$ is a finite subgroup of $$K, x\in F_1$$ and $$y_1,\ldots , y_m\in F_1\cap T$$. Thus $$y_1^{g_1}\cdots y_m^{g_m}\in (T\cap F_1)^G$$ and hence $$x\in (T\in F_1)^G\in F_1$$. It follows that
\begin{aligned} T^G\cap K&= \cup _{F\in \mathcal {L}}(T^G\cap F)\le \cup _{F\in \mathcal {L}}((T\cap F)^G\cap F) =\\&= \cup _{F\in \mathcal {L}}(T\cap F) = T\cap (\cup _{F\in \mathcal {L}}F) = T\cap K = T. \end{aligned}
Since $$T\le T^G\cap K,\, T = T^G\cap K$$ and then $$K$$ is normal sensitive in $$G$$. $$\square$$

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

By Lemma 1, every finite subgroup $$K$$ of $$P$$ is normal sensitive in $$G$$. Let $$H$$ be an arbitrary subgroup of $$K$$. Since $$K$$ is finite, $$H$$ is subnormal in $$K$$, that is there exists a series
\begin{aligned} H = H_0\trianglelefteq H_1\trianglelefteq \cdots \trianglelefteq H_{n-1}\trianglelefteq H_n = K. \end{aligned}
Since $$H_{n-1}$$ is normal sensitive in $$G$$ and $$H_{n-2}$$ is normal in $$H_{n-1}$$,
\begin{aligned} H_{n-2} = (H_{n-2})^G\cap H_{n-1}. \end{aligned}
On the other hand,
\begin{aligned} (H_{n-2})^K\cap H_{n-1}\le (H_{n-2})^G\cap H_{n-1}, \end{aligned}
and then $$(H_{n-2})^K\cap H_{n-1} = H_{n-2}$$. In particular, $$H_{n-2}$$ is the intersection of two normal subgroups of $$K$$ and then $$H_{n-2}$$ is normal in $$K$$. Proceeding in this way, after finitely many steps we obtain that $$H$$ is normal in $$K$$. Therefore $$K$$ is a Dedekind group and hence $$P$$ is a Dedekind group. Clearly, if $$p\ne 2,\, P$$ has to be abelian. $$\square$$

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

Let $$P$$ be a Sylow $$p$$-subgroup of $$F$$ so that $$P$$ is a Dedekind group by Lemma 2. If $$L = N_F(P)$$, then we have $$L = P\rtimes E$$ for some subgroup $$E$$. By Lemma 3, every subgroup of $$P$$ is $$L$$-invariant. Suppose that $$L\ne C_F(P)$$ and pick $$x\in L$$ such that $$x\notin C_F(P)$$ but $$x^q\in C_F(P)$$, where $$q$$ is a prime. Then there is some $$a\in P$$ such that $$a^x\ne a$$. Put $$A = \langle a\rangle$$ and let
\begin{aligned} A_j = \varOmega _j(A) = \{c\in A\ |\ c^{p_j}=1\} \end{aligned}
be the $$j$$th-layer of $$A$$. If $$x\in C_F(A_1)$$, then $$x\in C_F(A_{j+1}/A_j)$$ for each index $$j$$. It follows that $$\langle A,x\rangle$$ is nilpotent, which shows that $$x\in C_F(A)$$, a contradiction. Hence $$x\notin C_F(A_1)$$. Then $$q$$ divides $$p-1$$ and, in particular, $$q < p$$. This second contradiction proves the equation $$L = C_F(P)$$. Applying [2, Theorem 1], we obtain that the Sylow $$p'$$-subgroup of $$F$$ is normal in $$F$$, as required. $$\square$$
Let $${\mathfrak {X}}$$ be a class of groups. We recall that, if $$G$$ is a group, the intersection $$G_{\mathfrak {X}}$$ of all normal subgroups $$H$$ of $$G$$ such that $$G/H\in {\mathfrak {X}}$$ is called 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
\begin{aligned} G_{L{\mathfrak {N}}} = \bigcup _{F\in \mathcal {L}}F_{\mathfrak {N}}. \end{aligned}
The next result is crucial for this investigation.

### 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

Let $$K$$ be an arbitrary finite subgroup of $$G$$ and put $$D = K_{\mathfrak {N}}$$. As we remarked above, $$D\le L$$ so that every $$p$$-subgroup of $$G$$ is normal sensitive in $$G$$ for every $$p\in \pi (D)$$. Suppose that
\begin{aligned} \pi (D) = \{p_1,\ldots , p_m\}, \end{aligned}
where $$p_1 < \cdots < p_m$$. By Lemma 4, $$D$$ has a series of normal subgroups
\begin{aligned} \langle 1\rangle = D_{m+1}\le D_m\le \cdots \le D_2\le D_1= D, \end{aligned}
where $$D_1 = D_2\rtimes P_1$$ and $$P_1$$ is a Sylow $$p_1$$-subgroup of $$D$$, $$D_2 = D_3\rtimes P_2$$ and $$P_2$$ is a Sylow $$p_2$$-subgroup of $$D$$,..., and $$D_m = P_m$$ is a Sylow $$p_m$$-subgroup of $$D$$. Clearly $$P_m$$ is normal in $$K$$ and then every subgroup of $$D_m$$ is $$K$$-invariant by Lemma 3. Consider the factor $$D_{m-1}/D_m\cong P_{m-1}$$. Suppose first that $$p_{m-1}\ne 2$$. Put $$A/D_m = \varOmega _1(D_{m-1}/D_m)$$. By Lemma 2, a Sylow $$p_{m-1}$$-subgroup of $$K$$ is abelian. It follows that $$K/C_K(A/D_m)$$ is a $$p_{m-1}'$$-group. Applying Maschke’s theorem (see [10, Corollary 5.15] for example) we obtain that
\begin{aligned} A/D_m = B_1/D_m\times \cdots \times B_t/D_m, \end{aligned}
where each $$B_j/D_m$$ is a $$K$$-chief factor. Suppose that there exists a number $$s$$ such that $$|B_s/D_m|> p_{m-1}$$. Let $$x\in B_s{\setminus } D_m$$. There is no loss if we assume that $$|x| = p_{m-1}$$. Consider the subgroup $$X = \langle x\rangle ^K$$. Then $$Y = X\cap D_m$$ is a normal Sylow $$p_m$$-subgroup of $$X$$, and so $$X = Y\rtimes Q_2$$, where $$Q_2$$ is a Sylow $$p_{m-1}$$-subgroup of $$X$$. Without loss of generality we may assume that $$x\in Q_2$$. Since $$|Q_2| = |B_s/D_m|$$, $$\langle x\rangle \ne Q_2$$. Since $$Q_2$$ is normal sensitive in $$G$$ and $$\langle x\rangle$$ is normal in $$Q_2$$, $$\langle x\rangle = \langle x\rangle ^G\cap Q_2$$. On the other hand, $$\langle x\rangle ^K\cap Q_2\le \langle x\rangle ^G\cap Q_2$$, and then $$\langle x\rangle ^K\cap Q_2 = \langle x\rangle$$. But $$\langle x\rangle ^K\cap Q_2 = X\cap Q_2 = Q_2$$, and we obtain a contradiction. This contradiction shows that $$|B_j/D_m| = p_{m-1}$$ for every index $$1\le j\le t$$. Now it is not hard to see that every $$K$$-chief factor of $$D_{m-1}/D_m$$ has order $$p_{m-1}$$.

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

By Corollary 2, $$G$$ is locally nilpotent and therefore we may express
\begin{aligned} L = \text{ Dr }_{p\in \pi (L)}L_p, \end{aligned}
where $$L_p$$ is the Sylow $$p$$-subgroup of $$L$$ for each $$p\in \pi (L)$$. By Corollary 1, $$2\notin \pi (L)$$, and by Lemma 2, each $$L_p$$ is abelian. Therefore $$L$$ is abelian. Moreover Lemma 3 shows that every subgroup of each $$L_p$$ is $$G$$-invariant, and it follows that every subgroup of $$L$$ is $$G$$-invariant. $$\square$$

### 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

By Corollary 3, $$L$$ is abelian and so we may express
\begin{aligned} L = \text{ Dr }_{p\in \pi (L)}L_p, \end{aligned}
where $$L_p$$ is the Sylow $$p$$-subgroup of $$L$$. By Corollary 1, $$2\notin \pi (L)$$, and then, by Lemma 2, the Sylow $$p$$-subgroups of $$G$$ are abelian for every $$p\in \pi (L)$$. It follows that $$G/C_G(L_p)$$ is a $$p$$-group. Since every subgroup of $$L_p$$ is $$G$$-invariant by Corollary 3, $$G/C_G(L_p)$$ is a cyclic group of order dividing $$p-1$$ (see [13, Theorem 1.5.6] for example). Put
\begin{aligned} Q = \text{ Dr }_{p\ne q\in \pi (L)}L_q. \end{aligned}
Let $$H$$ be a non-identity subgroup of $$L_p$$ so that $$H$$ is normal in $$G$$ and $$C_G(L_p)\le C_G(H)$$. Applying [4, Proposition 2.12], we obtain a direct decomposition
\begin{aligned} H = C_H(G)\times [H, G]. \end{aligned}
If we suppose that $$H\ne [H,G]$$, then $$C_H(G)\ne \langle 1\rangle$$. Therefore $$C_{L_p}(G)\ne \langle 1\rangle$$. Successive applications of [4, Proposition 2.12] show that $$L_p\ne [L_p,G]$$. Put
\begin{aligned} R = [L_p,G]\times Q \end{aligned}
so that $$L\ne R,\, [L,G]\le R$$ and $$L/R\le \zeta (G/R)$$. Since $$G/L$$ is locally nilpotent, $$G/R$$ is also locally nilpotent, which contradicts the choice of $$L$$. This contradiction proves that $$H = [H,G]$$. Let now $$H$$ be a non-identity subgroup of $$L$$. Then
\begin{aligned} H = \text{ Dr }_{p\in \pi (H)}H_p, \end{aligned}
where $$H_p$$ is the Sylow $$p$$-subgroup of $$H$$. We proved above that $$[H_p,G] = H_p$$ for every $$p\in \pi (H)$$. Therefore $$[H,G] = H$$, as required. $$\square$$

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

Suppose the contrary and pick $$p\in \pi (L)\cap \pi (G/L)$$. By Corollary 1, $$p\ne 2$$. By Corollary 3, $$L$$ is abelian, so that $$L = P\times Q$$, where $$P$$ is the Sylow $$p$$-subgroup of $$L$$ and $$Q$$ is the Sylow $$p'$$-subgroup of $$L$$. By Lemma 2, the Sylow $$p$$-subgroups of $$G$$ are abelian. Let $$S/L$$ be the Sylow $$p$$-subgroup of $$G/L$$. Pick $$xL, yL\in S/L$$ and put $$X/L = \langle xL,yL\rangle$$. Since $$G$$ is locally finite, $$X/L$$ is finite. Then $$X = YL$$ for some finite subgroup $$Y$$. Let $$Z$$ be the Sylow $$p$$-subgroup of $$Y$$. Then $$Z(Y\cap L)/(Y\cap L)$$ is a Sylow $$p$$-subgroup of $$Y/(Y\cap L)$$. But $$Y/(Y\cap L)\cong YL/L = X/L$$ is a $$p$$-group so $$Z(Y\cap L)/(Y\cap L) = Y/(Y\cap L)\cong X/L$$. In particular, $$X/L$$ is abelian and so is $$S/L$$. Since the factor-group $$G/L$$ is locally nilpotent, we may express
\begin{aligned} G/L = S/L\times R/L, \end{aligned}
where $$R/L$$ is the Sylow $$p'$$-subgroup of $$G/L$$. Since $$S/L$$ is abelian, $$S/L\le \zeta (G/L)$$. Pick a non-identity element $$aL\in S/L$$ such that $$a^p\in L$$, and put $$A = \langle a,L\rangle$$. As other times, $$G/C_G(P)$$ is a cyclic group of order dividing $$p-1$$ and then we may choose a $$p'$$-element $$v$$ such that $$G = \langle v\rangle C_G(P)$$. By Corollary 3 and Corollary 4, every non-identity subgroup $$H$$ of $$P$$ is $$\langle v\rangle$$-invariant and $$[H,v] = H$$. Since $$v\notin L,\, [HQ/Q,vQ] = HQ/Q$$. Clearly $$A/Q$$ is a normal $$p$$-subgroup of $$G/Q$$ and $$[A/Q,G/Q] = L/Q$$. Applying [4, Proposition 2.12], we obtain a direct decomposition
\begin{aligned} A/Q = C_{A/Q}(vQ)\times [A/Q,vQ] = C_{A/Q}(vQ)\times L/Q. \end{aligned}
Since $$|A/L| = p$$, $$C_{A/Q}(vQ) = \langle cQ\rangle$$. Moreover, without loss of generality we may suppose that $$|c| = p$$. Then $$d := [c,v]\in Q$$. It follows that the cyclic subgroup $$\langle d\rangle$$ is $$G$$-invariant. Pick $$b\in P$$ of order $$p$$ and put $$B = \langle b,c,v\rangle$$. Then $$B = K\langle v\rangle$$, where $$K = \langle d\rangle \rtimes (\langle b\rangle \times \langle c\rangle )$$. We remarked above that $$[\langle b\rangle ,v] = \langle b\rangle$$ and so $$b^v = b^k$$, where $$k$$ is a $$p'$$-number such that $$k\not \equiv 1 (\text{ mod }\ p)$$. We have $$(bc)^v = b^vc^v = b^kcd$$, which implies that $$1\ne b^{k-1}d\in \langle bc\rangle ^{\langle v\rangle }$$. Since $$(|d|,p) = 1$$, $$\langle b^{k-1}\rangle = \langle b\rangle$$ is a Sylow $$p$$-subgroup of $$\langle b^{k-1}d\rangle \le \langle bc\rangle ^{\langle v\rangle }$$. Thus $$b\in \langle bc\rangle ^{\langle v\rangle }$$ and hence $$c\in \langle bc\rangle ^{\langle v\rangle }$$ and $$\langle bc\rangle ^{\langle v\rangle } = K$$. Being a $$p$$-subgroup, $$\langle b\rangle \times \langle c\rangle$$ is normal sensitive in $$G$$. Since $$\langle bc\rangle$$ is normal in $$\langle b\rangle \times \langle c\rangle$$,
\begin{aligned} \langle bc\rangle ^G\cap (\langle b\rangle \times \langle c\rangle ) = \langle bc\rangle . \end{aligned}
Then
\begin{aligned} \langle bc\rangle ^{\langle v\rangle }\cap (\langle b\rangle \times \langle c\rangle )\le \langle bc\rangle ^G\cap (\langle b\rangle \times \langle c\rangle ) \end{aligned}
and then
\begin{aligned} \langle bc\rangle = \langle bc\rangle ^{\langle v\rangle }\cap (\langle b\rangle \times \langle c\rangle ). \end{aligned}
On the other hand,
\begin{aligned} <\langle bc\rangle ^{\langle v\rangle }\cap (\langle b\rangle \times \langle c\rangle ) = K\cap (\langle b\rangle \times \langle c\rangle ) = \langle b\rangle \times \langle c\rangle \end{aligned}
and we obtain a contradiction. This contradiction shows the result.$$\square$$

## 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

As we mentioned in the Introduction, (ii) and (iii) are equivalent (see [9]). Suppose that every primary subgroup of $$G$$ is normal sensitive in $$G$$. In particular, every $$p$$-subgroup of $$G$$ is normal sensitive in $$G$$ for every $$p\in \pi (L)$$, where $$L$$ is the locally nilpotent residual of $$G$$. By Theorem A, $$G$$ satisfies the conditions (i)–(v) of its statement. Let $$p\in \pi (G/L)$$. By Lemma 2, every Sylow $$p$$-subgroup of $$G$$ is a Dedekind group. Let $$S/L$$ be the Sylow $$p$$-subgroup of $$G/L$$, and pick $$xL, yL\in S/L$$. Put $$X/L = \langle xL,yL\rangle$$. Since $$G$$ is locally finite, $$X/L$$ is finite. Then $$X = YL$$, for some finite subgroup $$Y$$. Let $$Z$$ be the Sylow $$p$$-subgroup of $$Y$$. Then $$Z(Y\cap L)/(Y\cap L)$$ is a Sylow $$p$$-subgroup of $$Y/(Y\cap L)$$. Since $$Y/(Y\cap L)\cong YL/L = X/L$$ is a $$p$$-group, we have that
\begin{aligned} Z(Y\cap L)/(Y\cap L) = Y/(Y\cap L)\cong X/L. \end{aligned}
Since $$Z$$ is a Dedekind group, $$X/L$$ is also a Dedekind group. It follows that $$(xL)^{yL} = (xL)^k$$ for some integer $$k$$. This holds for every $$yL\in S/L$$ so that $$\langle xL\rangle$$ is normal in $$S/L$$. Therefore $$S/L$$ is Dedekind group. By [11, Lemma 5.2.2 and Theorem 6.1.1], $$G$$ is a $$\overline{T}$$-group. Thus we obtain that (i) implies (iii). As we remarked above, (iii) implies (ii) and (ii) implies (i).$$\square$$

## 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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