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Monatshefte für Mathematik

, Volume 175, Issue 2, pp 175–185 | Cite as

Groups whose primary subgroups are normal sensitive

  • Adolfo Ballester-Bolinches
  • Leonid A. Kurdachenko
  • Javier Otal
  • Tatiana Pedraza
Article
  • 126 Downloads

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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Copyright information

© Springer-Verlag Wien 2013

Authors and Affiliations

  • Adolfo Ballester-Bolinches
    • 1
  • Leonid A. Kurdachenko
    • 2
  • Javier Otal
    • 3
  • Tatiana Pedraza
    • 4
  1. 1.Departament d’ÀlgebraUniversitat de València Burjassot (València)Spain
  2. 2.Department of AlgebraNational University of DnepropetrovskDnepropetrovskUkraine
  3. 3.Departmento de Matemáticas, IUMAUniversidad de Zaragoza Zaragoza Spain
  4. 4.Instituto Universitario de Matemática Pura y AplicadaUniversidad Politécnica de Valencia ValenciaSpain

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