$c^*$-Normal and s-semipermutable subgroups in finite groups

Abstract

A subgroup $H$ of a group $G$ is called $c^*$-normal in $G$ if there exists a normal subgroup $N$ of $G$ such that $G=HN$ and $H\cap N$ is $S$-quasinormally embedded in $G$. A subgroup $K$ of $G$ is said to be $s$-semipermutable if it is permutable with every Sylow $p$-subgroup of $G$ with $(p, |K|)=1$. In this article, we investigate the influence of $c^*$-normality and $s$-semipermutability of subgroups on the structure of finite groups and generalize some known results.

1 Introduction

Throughout only finite groups are considered. Terminologies and notations employed agree with standard usage, as in Robinson [8].

Two subgroups $H$ and $K$ of a group $G$ are said to be permutable if $HK = KH$. The subgroup $H$ is said to be $S$-quasinormal in $G$ if $H$ permutes with every Sylow subgroups of $G$, i.e., $HP = PH$ for any Sylow subgroup $P$ of $G$. This concept was introduced by Kegel in [7] and has been studied widely by many authors, such as [2, 9]. Recently, There is a generalization of $S$-quasinormality in [14]. The subgroup $H$ is called $s$-semipermutable in $G$ if $H$ permutes with every Sylow $p$-subgroup of $G$ with $(|H |, p )=1$. An $s$-semipermutable subgroup is no need to be an $S$-quasinormal subgroup. $S_3$ is a counter-example. On the other hand, Wang [10] introduced the concept of $c$-normal subgroups. The subgroup $H$ is said to be $c$-normal in $G$ if there exists a normal subgroup $U$ of $G$ such that $G= HU$ and $H\cap U$ is contained in $H_G$, where $H_G$ is the maximal normal subgroup of $G$ which is contained in $H$. The $c$-normality is a generalization of the normality. Applying the $c$-normality of subgroups, Wang obtained new criteria for supersolvability of groups. In 2007, Wei and Wang [11] introduced the concept of $c^*$-normal subgroups which is both $c$-normality and $S$-quasinormal embedding and used the $c^*$-normality of maximal subgroups to give some necessary and sufficient conditions for a group to be $p$-nilpotent, $p$-supersolvable or supersolvable. Based on the observation above concepts, we note that $c^*$-normal subgroups and $s$-semipermutable subgroups are two different concepts. There are examples to show that $s$-semipermutable subgroups are not $c^*$-normal subgroups and in general the converse is also false. In this paper, we investigate $s$-semipermutable and $c^*$-normal subgroups of $G$ and give criteria for a group belonging to $\fancyscript{F}$. Some interesting results are obtained and known results on this topic are generalized.

2 Preliminaries

Lemma 2.1

[11, Lemma 2.3] Let $H$ be a subgroup of a group $G$.

• (1) If $H$ is $c^*$-normal in $G$ and $H\le M\le G$, then $H$ is $c^*$-normal in $M$.

• (2) Let $N\lhd G$ and $N \le H$. Then $H$ is $c^*$-normal in $G$ if and only if $H{/}N$ is $c^*$-normal in $G{/}N$.

• (3) Let $\pi$ be a set of primes, $H$ a $\pi$-subgroup of $G$ and $N$ a normal $\pi ^\prime$-subgroup of $G$. If $H$ is $c^*$-normal in $G$, then $HN{/}N$ is $c^*$-normal in $G{/}N$.

Lemma 2.2

[14, Property] Suppose that $H$ is an $s$-semipermutable subgroup of $G$. Then

• (1) If $H \le K\le G$, then $H$ is $s$-semipermutable in $K$.

• (2) Let $N$ be a normal subgroup of $G$. If $H$ is a $p$-group for some prime $p\in \pi (G)$, then $HN{/}N$ is $s$-semipermutable in $G{/}N$.

• (3) If $H\le O_p(G)$, then $H$ is $S$-quasinormal in $G$.

Lemma 2.3

[9, 11] Suppose that $U$ is $S$-quasinormally embedded in a group $G$, and that $H\le G$ and $K \unlhd G$.

• (1) If $U\le H$, then $U$ is $S$-quasinormally embedded in $H$.

• (2) $UK$ is $S$-quasinormally embedded in $G$ and $UK{/}K$ is $S$-quasinormally embedded in $G{/} K$.

• (3) If $K\le H$ and $H{/} K$ is $S$-quasinormally embedded in $G{/}K$, then $H$ is $S$-quasinormally embedded in $G$.

• (4) A $p$-subgroup $H$ of $G$ is $S$-quasinormal in $G$ if and only if $N_G(H)\ge O^p(G)$ for some prime $p\in \pi (G).$

Lemma 2.4

[11, Lemma 2.8] Let $G$ be a group and let $p$ be a prime number dividing $| G|$ with $( | G|,p -1) = 1$.Then

• (1) If $N$ is normal in $G$ of order $p$, then $N$ lies in $Z( G)$;

• (2) If $G$ has cyclic Sylow $p$-subgroups, then $G$ is $p$-nilpotent;

• (3) If $M$ is a subgroup of $G$ with index $p$, then $M$ is normal in $G$.

Lemma 2.5

[1, A, Lemma 1.2] Let $U , V$ and $W$ be subgroups of a group $G$. The following statements are equivalent.

• (1) $U\cap V W =(U\cap V )( U\cap W )$.

• (2) $U V\cap U W = U (V \cap W )$.

Lemma 2.6

[4, 6.4.8] Let $H$, $K$ be subgroups of the group $G$ such that

$$\begin{aligned} (|G : H |, |G : K |)=1. \end{aligned}$$

Then $G = HK$ and $|G : H\cap K | = |G : H || G : K |$.

Lemma 2.7

[11, Lemma 2.5] Let $G$ be a group, $K$ an $S$-quasinormal subgroup of $G$ and $P$ a Sylow $p$-subgroup of $K$, where $p$ is a prime. If $K_G=1$, then $P$ is $S$-quasinormal in $G$.

Lemma 2.8

[11, Theorem 4.1] Let $\fancyscript{F}$ be a saturated formation containing $\fancyscript{U}$. Suppose that $G$ is a group with a normal subgroup $H$ such that $G{/} H\in \fancyscript{F}$. If all maximal subgroups of any Sylow subgroup of $H$ are $c^*$-normal in $G$, then $G\in \fancyscript{F}$.

Lemma 2.9

[11, Theorem 4.3] Let $\fancyscript{F}$ be a saturated formation containing $\fancyscript{U}$. Suppose that $G$ is a group with a normal subgroup $H$ such that $G{/} H\in \fancyscript{F}$. If all maxim al subgroups of any Sylow subgroup of $\fancyscript{F}^*(H)$ are $c^*$-normal in $G$, then $G\in \fancyscript{F}$.

Lemma 2.10

[5, X, 13] Let $G$ be a group. If $F^*(G)$ is solvable, then $F^*(G)= F (G).$

Lemma 2.11

[5, IV, Satz 4.7] If $P$ is a Sylow $p$-subgroup of $G$ and $N\unlhd G$ such that $P\cap N\le \Phi (P )$, then $N$ is $p$-nilpotent.

3 Main results

Theorem 3.1

Let $G$ be a group and $P$ a Sylow $p$-subgroup of $G$, where $p$ is the smallest prime dividing $|G|$. If every maximal subgroup of $P$ is either $s$-semipermutable or $c^*$-normal in $G$, then $G$ is $p$-nilpotent.

Proof

Asuume that the theorem is false and $G$ is a counterexample with minimal order. We will consider the following steps.

• (1) $G$ has a unique minimal normal subgroup $N$ such that $G{/}N$ is $p$-nilpotent and $\Phi ( G)=1$.

  Let $N$ be a minimal normal subgroup of $G$. We have to show $G{/}N$ satisfies the hypotheses of the theorem. Let $M{/}N$ be a maximal subgroup of $PN{/}N$. We can see $M = P_1 N$ for some maximal subgroup $P_1$ of $P$. By the hypotheses, $P_1$ is either $s$-semipermutable or $c^*$-normal in $G$. If $P_1$ is $c^*$-normal in $G$, then there is a normal subgroup $K_1$ of $G$ such that $G = P_1 K_1$ and $P_1\cap K_1$ is $S$-quasinormally embedded in $G$. Then $G{/}N= M{/}N\cdot K_1 N{/}N= P_1 N{/}N\cdot K_1 N{/}N$. It is easy to see that $K_1 N{/}N$ is normal in $G{/}N$. Since $(|N : P_1\cap N |, |N : K_1\cap N |) = 1, (P_1\cap N )( K_1\cap N ) = N = N \cap G = N\cap (P_1 K_1)$ by Lemma 2.6. Now using Lemma 2.5, $( P_1 N )\cap (K_1 N ) = ( P_1\cap K_1)N$. It follows that $( P_1 N ){/}N\cap (K_1 N ){/}N =(P_1\cap K_1)N{/}N$ is $S$-quasinormally embedded in $G{/}N$ by Lemma 2.3. Thus $M{/}N$ is $c^*$-normal in $G{/}N$. If $P_1$ is $s$-semipermutable in $G$, then $M{/}N= P_1 N{/}N$ is $s$-semipermutable in $G{/}N$ by Lemma 2.2. Consequently, $G{/}N$ satisfies the hypotheses of the theorem. The choice of $G$ yields that $G{/}N$ is $p$-nilpotent. The uniqueness of $N$ and $\Phi (G)=1$ are obvious.

• (2) $O_{p^\prime }(G)=1$.

  If $O_{p^\prime }(G)\ne 1$, then $N \le O_{p^\prime }(G)$ by (1). By Lemmas 2.1 and 2.2, $G{/}N$ satisfies the hypotheses, hence $G{/}N$ is $p$-nilpotent. Now the $p$-nilpotency of $G{/}N$ implies the $p$-nilpotency of $G$, a contradiction.

• (3) $O_p(G)=1$ and $G$ is not solvable. paraIf $O_p(G)\ne 1$, (1) yields $N\le O_p(G)$ and $\Phi ( O_p(G))\le \Phi (G)=1$. Hence, $G$ has a maximal subgroup $M$ such that $G = M N$ and $M\cap N = 1$. Since $O_p(G)\cap M$ is normalized by $N$ and $M$, $O_p(G)\cap M$ is normal in $G$. The uniqueness of $N$ yields $N = O_p(G)$. Obviously $P = N M_p = N (P\cap M )$. Since $P\cap M < P$, we can take a maximal subgroup $P_1$ of $P$ such that $P\cap M\le P_1<\cdot P$. Then $P = N P_1$ and $P\cap M= P_1\cap M$. By the hypotheses of theorem, $P_1$ is either $s$-semipermutable or $c^*$-normal in $G$. If $P_1$ is $c^*$-normal in $G$, then there is a normal subgroup $K_1$ such that $G = P_1 K_1$ and $P_1\cap K_1$ is $S$-quasinormally embedded in $G$. Thus $P_1\cap K_1$ is a Sylow $p$-subgroup of some $S$-quasinormal subgroup $K$ of $G$. If $K_G \ne 1$, by (1) we have that $N\le K_G$. Hence, $P=NP_1\le P_1$, a contradiction. If $K_G = 1$, by Lemma 2.7 we have that $P_1\cap K_1$ is $S$-quasinormal in $G$. It follows that $P_1\cap K_1$ is normalized by $P$ and $O^p(G)$. Now we know $P_1\cap K_1\lhd G$. If $(P_1\cap K_1)M=G$, then $P_1M=PM=G$ and so $P_1 = P$, a contradiction. Thus $(P_1\cap K_1)M=M$ and so $(P_1\cap K_1)\le M$. On the other hand, $P_1\cap K_1\le N$. We know $P_1\cap K_1\le N\cap M$. Hence, $P_1\cap K_1=1$ and so $|P\cap K_1|=p$. So $|K|_p= p$. By the uniqueness of $N$, we have that $N\le K_1$, of course, $N$ is a cyclic group of order $p$. By Lemma 2.4, $N\le Z(G)$. Since $G{/}N$ is $p$-nilpotent, $G$ is also $p$-nilpotent, a contradiction. Now suppose that $P_1$ is $s$-semipermutable in $G$. Then $P_1 M_q$ is a group for $q\ne p$. Therefore, $P_1<M_p , M_q|q\in \pi (M ), q\ne p>= P_1 M$ is a group. Then $P_1 M = M$ or $G$ by maximality of $M$. If $P_1 M = G$, then $P = P\cap P_1 M = P_1(P \cap M ) = P_1$, which is a contradiction. If $P_1 M = M$, then $P_1\le M$. Hence, $P_1\cap N = 1$ and $N$ is of prime order. Then the $p$-nilpotency of $G{/}N$ implies the $p$-nilpotency of $G$, a contradiction. Combining this with (2), it is easy to see that $G$ is not solvable, now thus (3) holds.

• (4) For any $q\ne p$, $P G_q < G$, where $G_q$ is a Sylow $q$-subgroup of $G$. That is to say, $P G_q$ is $p$-nilpotent.

  At first, we have $NP=G$. In fact, if $NP< G$, then $NP$ is $p$-nilpotent since $NP$ satisfies the hypotheses of theorem. Hence, $N$ is $p$-nilpotent and by (1) we know $N$ is a nontrivial $p$-group, but this is a contradiction with (3). So we have that $NP=G$. If for all $P_1<\cdot P$, we have that $NP_1< G$. Then $(P\cap N)P_1<P$ and so $P\cap N\le P_1$. Hence, $P\cap N\le \Phi (P)$ and $N$ is $p$-nilpotent by Lemma 2.11, a contradiction. So there exists $P_1<\cdot P$ such that $G=NP_1$. By the hypotheses, if $P_1$ is $c^*$-normal in $G$, then there is a normal subgroup $K$ such that $G = P_1 K_1$ and $P_1\cap K_1$ is $S$-quasinormally embedded in $G$. So $P_1\cap K_1 \in Syl_p(K)$, where $K$ is $S$-quasinormal in $G$. If $K_G\ne 1$, then $N\le K_G$. It follows that $P_1\cap K_1\cap N\in Syl_p(N)$. Now by $G=NP_1$ we get $P_1\in Syl_p(G)$, a contradiction. So $K_G=1$. By Lemma 2.7 we have that $P_1\cap K_1$ is $S$-quasinormal in $G$, so $P_1\cap K_1 \le O_p(G)=1$ and $P_1\cap K_1=1$. Moreover, $|P\cap K_1|=p$ and so $|K_1|_p=p$. By Lemma 2.4 we know $K_1$ is $p$-nilpotent. Of course, $N$ is also $p$-nilpotent, a contradiction. From [5, IV, Satz 2.8], it follows that $P$ is non-cyclic. We could take a maximal subgroup $P_2$ of $P$ satisfying $G=NP_2$. By the same argument, we know that $P_2$ cannot be $c^*$-normal in $G$. Now suppose that $P_i$ is $s$-semipermutable in $G$ and $P_iG_q$ is a group, $i = 1, 2$, where $G_q$ is a Sylow $q$-subgroup of $G$. Thus we have $P_1, P_2$ such that $P= P_1 P_2$. Hence $P G_q$ is a group. By (3) and the famous $p^aq^b$-theorem we infer $P G_q$ is a proper subgroup of $G$. Therefore, $P G_q$ is $p$-nilpotent by the minimality of $G$.

• (5) The final contradiction.

By (4) we have $[P , G_q]\le G_q$ for any $q\ne p$. Suppose that $S_1$ is an arbitrary subgroup of $P$. Let $N_G (S_1 ) = N_1$. Since $[ S_1, (N_1)_q]\le S_1\cap G_q= 1, S_1$ is centralized by $(N_1)_{p^\prime }$. Thus $G$ is $p$-nilpotent by the famous Frobenius Theorem [8, 10.3.2], which is the final contradiction. $\square$

Corollary 3.2

Suppose that $G$ is a group and $P$ a Sylow subgroup of $G$. If every maximal subgroup of $P$ is either $s$-semipermutable or $c^*$-normal in $G$, then $G$ has a Sylow tower of supersolvable type.

Proof

Let $p$ be the smallest prime dividing $|G|$ and $P$ a Sylow $p$-subgroup of $G$. By hypothesis, every maximal subgroup of $P$ is either $s$-semipermutable or $c^*$-normal in $G$. In particular, $G$ satisfies the condition of Theorem 3.1, so $G$ is $p$-nilpotent. Let $U$ be the normal $p$-complement of $G$. By Lemmas 2.1 and 2.2, $U$ satisfies the hypothesis. It follows by induction that $U$, and hence $G$ possesses the Sylow tower property of supersolvable type. $\square$

Corollary 3.3

[6, Theorem 3.1] Let $G$ be a group and $P = G_p$ a Sylow $p$-subgroup of $G$, where $p$ is the smallest prime dividing $|G|$. If every maximal subgroup of $P$ is either $s$-semipermutable or $c$-normal in $G$, then $G$ is $p$-nilpotent.

Corollary 3.4

Suppose that $G$ is a group and $P$ a Sylow subgroup of $G$. If every maximal subgroup of $P$ is either $s$-semipermutable or $c$-normal in $G$, then $G$ has a Sylow tower of supersolvable type.

We are now in a position to unify and generalize Theorem 4.1 and Theorem 4.3 in [11].

Theorem 3.5

Let $\fancyscript{F}$ be a saturated formation containing $\fancyscript{U}$. Suppose that $G$ is a group with a normal subgroup $H$ such that $G{/} H\in \fancyscript{F}$. If all maximal subgroups of any Sylow subgroup of $H$ are either $c^*$-normal or $s$-semipermutable in $G$, then $G\in \fancyscript{F}$.

Proof

Suppose that $P$ is a Sylow $p$-subgroup of $H$ , $\forall p\in \pi (H )$. Since every maximal subgroups of $P$ are either $c^*$-normal or $s$-semipermutable in $G$, thus in $H$ by Lemmas 2.1 and 2.2. By Corollary 3.2 we konw that $H$ has a Sylow tower of supersolvable type. Let $q$ be the maximal prime divisor of $|H |$ and $Q\in Syl_q(H )$. Then $Q$ char $H\unlhd G$. Since $( G{/}Q, H{/}Q)$ satisfies the hypotheses of the theorem, by induction, $G{/}Q\in \fancyscript{F}.$ Since $Q \le O_q(G)$, every maximal subgroups of $Q$ are either $S$-quasinormal or $c^*$-normal in $G$ by Lemma 2.2, in particular, $c^*$-normal in $G$. So $G\in \fancyscript{F}$ by Lemma 2.8. $\square$

Theorem 3.6

Let $\fancyscript{F}$ be a saturated formation containing $\fancyscript{U}$. Suppose that $G$ is a group with a normal subgroup $H$ such that $G{/} H\in \fancyscript{F}$. If all maxim al subgroups of any Sylow subgroup of $F^*(H)$ are either $c^*$-normal or $s$-semipermutable in $G$, then $G\in \fancyscript{F}$.

Proof

Suppose that $P$ is a Sylow $p$-subgroup of $F^*(H)$ , $\forall p\in \pi (F^*(H) )$. Since every maximal subgroups of $P$ are either $c^*$-normal or $s$-semipermutable in $G$, thus in $F^*(H)$ by Lemmas 2.1 and 2.2. By Corollary 3.2 we konw that $F^*(H)$ has a Sylow tower of supersolvable type, in particular, $F^*(H)$ is Solvable. By Lemma 2.10 we have that $F^*(H)=F(H)$. Since $F^*(H)_p=F(H)_p$, $\forall p\in (H)$, every maximal subgroups of $P$ are either $c^*$-normal or $S$-quasinormal in $G$ by Lemma 2.2, in particular, $c^*$-normal in $G$. Applying Lemma 2.9 we get that $G\in \fancyscript{F}$. $\square$

Corollary 3.7

[12, Theorem 3.1] Let $\fancyscript{F}$ be a saturated formation containing $\fancyscript{U}$. Suppose that $G$ is a group with a normal subgroup $H$ such that $G{/} H\in \fancyscript{F}$. If all maximal subgroups of every Sylow subgroup of $F^*(H)$ are c-normal in $G$, then $G\in \fancyscript{F}$.

Corollary 3.8

[14, Theorem 1] Let $\fancyscript{F}$ be a saturated formation containing $\fancyscript{U}$. Suppose that $G$ is a group with a normal subgroup $H$ such that $G{/} H\in \fancyscript{F}$. If all maximal subgroups of any Sylow subgroup of $H$ are $s$-semipermutable in $G$, then $G\in \fancyscript{F}$.

Corollary 3.9

[14, Theorem 2] Let $\fancyscript{F}$ be a saturated formation containing $\fancyscript{U}$. Suppose that $G$ is a group with a normal subgroup $H$ such that $G{/} H\in \fancyscript{F}$. If all maximal subgroups of any Sylow subgroup of $F^*(H)$ are $s$-semipermutable in $G$, then $G\in \fancyscript{F}$.