Subnormality and residuals for saturated formations: A generalization of Schenkman’s theorem

Let 𝐺 be a finite group, and let 𝔉 be a hereditary saturated formation. We denote by Z F ⁢ ( G ) \mathbf{Z}_{\mathfrak{F}}(G) the product of all normal subgroups 𝑁 of 𝐺 such that every chief factor H / K H/K of 𝐺 below 𝑁 is 𝔉-central in 𝐺, that is, ( H / K ) ⋊ ( G / C G ⁢ ( H / K ) ) ∈ F (H/K)\rtimes(G/\mathbf{C}_{G}(H/K))\in\mathfrak{F} . A subgroup A ⩽ G A\leqslant G is said to be 𝔉-subnormal in the sense of Kegel, or 𝐾-𝔉-subnormal in 𝐺, if there is a subgroup chain A = A 0 ⩽ A 1 ⩽ ⋯ ⩽ A n = G A=A_{0}\leqslant A_{1}\leqslant\cdots\leqslant A_{n}=G such that either A i - 1 ⁢ ⊴ ⁢ A i A_{i-1}\trianglelefteq A_{i} or A i / ( A i - 1 ) A i ∈ F A_{i}/(A_{i-1})_{A_{i}}\in\mathfrak{F} for all i = 1 , … , n i=1,\ldots,n . In this paper, we prove the following generalization of Schenkman’s theorem on the centraliser of the nilpotent residual of a subnormal subgroup: Let 𝔉 be a hereditary saturated formation containing all nilpotent groups, and let 𝑆 be a 𝐾-𝔉-subnormal subgroup of 𝐺. If Z F ⁢ ( E ) = 1 \mathbf{Z}_{\mathfrak{F}}(E)=1 for every subgroup 𝐸 of 𝐺 such that S ⩽ E S\leqslant E , then C G ⁢ ( D ) ⩽ D \mathbf{C}_{G}(D)\leqslant D , where D = S F D=S^{\mathfrak{F}} is the 𝔉-residual of 𝑆.

S-semipermutability of subgroups of p-nilpotent residual and p-supersolubility of a finite group

A subgroup [Formula: see text] of a finite group [Formula: see text] is said to be [Formula: see text]-semipermutable in [Formula: see text], if [Formula: see text] permutes with all Sylow [Formula: see text]-subgroups of [Formula: see text] for the primes [Formula: see text] not dividing [Formula: see text]. In this paper, we consider the [Formula: see text]-semipermutability of some [Formula: see text]-subgroups to investigate the [Formula: see text]-supersolubility of a finite group. Some interesting results are obtained which extend some known results.

ON THE σ-NILPOTENT NORM AND THE σ-NILPOTENT LENGTH OF A FINITE GROUP

Let G be a finite group and σ = {σi| i ∈ I} some partition of the set of all primes $\Bbb{P}$ . Then G is said to be: σ-primary if G is a σi-group for some i; σ-nilpotent if G = G1× … × Gt for some σ-primary groups G1, … , Gt; σ-soluble if every chief factor of G is σ-primary. We use $G^{{\mathfrak{N}}_{\sigma}}$ to denote the σ-nilpotent residual of G, that is, the intersection of all normal subgroups N of G with σ-nilpotent quotient G/N. If G is σ-soluble, then the σ-nilpotent length (denoted by lσ (G)) of G is the length of the shortest normal chain of G with σ-nilpotent factors. Let Nσ (G) be the intersection of the normalizers of the σ-nilpotent residuals of all subgroups of G, that is, $${N_\sigma }(G) = \bigcap\limits_{H \le G} {{N_G}} ({H^{{_\sigma }}}).$$ Then the subgroup Nσ (G) is called the σ-nilpotent norm of G. We study the relationship of the σ-nilpotent length with the σ-nilpotent norm of G. In particular, we prove that the σ-nilpotent length of a σ-soluble group G is at most r (r > 1) if and only if lσ (G/ Nσ (G)) ≤ r.

Nilpotent residual and fitting subgroup of fixed points in finite groups

Let q be a prime and A a finite q-group of exponent q acting by automorphisms on a finite {q^{\prime}} -group G. Assume that A has order at least {q^{3}} . We show that if {\gamma_{\infty}(C_{G}(a))} has order at most m for any {a\in A^{\#}} , then the order of {\gamma_{\infty}(G)} is bounded solely in terms of m. If the Fitting subgroup of {C_{G}(a)} has index at most m for any {a\in A^{\#}} , then the second Fitting subgroup of G has index bounded solely in terms of m.

Finite groups with two supersoluble subgroups

LetGbe a finite group. In this paper we obtain some sufficient conditions for the supersolubility ofGwith two supersoluble non-conjugate subgroupsHandKof prime index, not necessarily distinct. It is established that the supersoluble residual of such a group coincides with the nilpotent residual of the derived subgroup. We prove thatGis supersoluble in the following cases: one of the subgroupsHorKis nilpotent; the derived subgroup{G^{\prime}}ofGis nilpotent;{|G:H|=q>r=|G:K|}andHis normal inG. Also the supersolubility ofGwith two non-conjugate maximal subgroupsMandVis obtained in the following cases: all Sylow subgroups ofMand ofVare seminormal inG; all maximal subgroups ofMand ofVare seminormal inG.

FITTING SUBGROUP AND NILPOTENT RESIDUAL OF FIXED POINTS

Let $q$ be a prime and let $A$ be an elementary abelian group of order at least $q^{3}$ acting by automorphisms on a finite $q^{\prime }$-group $G$. We prove that if $|\unicode[STIX]{x1D6FE}_{\infty }(C_{G}(a))|\leq m$ for any $a\in A^{\#}$, then the order of $\unicode[STIX]{x1D6FE}_{\infty }(G)$ is $m$-bounded. If $F(C_{G}(a))$ has index at most $m$ in $C_{G}(a)$ for any $a\in A^{\#}$, then the index of $F_{2}(G)$ is $m$-bounded.

A sufficient condition for nilpotency of the nilpotent residual of a finite group

LetGbe a finite group with the property that if{a,b}are powers of{\delta_{1}^{*}}-commutators such that{(|a|,|b|)=1}, then{|ab|=|a||b|}. We show that{\gamma_{\infty}(G)}is nilpotent.

Almost Engel finite and profinite groups

Let [Formula: see text] be an element of a group [Formula: see text]. For a positive integer [Formula: see text], let [Formula: see text] be the subgroup generated by all commutators [Formula: see text] over [Formula: see text], where [Formula: see text] is repeated [Formula: see text] times. We prove that if [Formula: see text] is a profinite group such that for every [Formula: see text] there is [Formula: see text] such that [Formula: see text] is finite, then [Formula: see text] has a finite normal subgroup [Formula: see text] such that [Formula: see text] is locally nilpotent. The proof uses the Wilson–Zelmanov theorem saying that Engel profinite groups are locally nilpotent. In the case of a finite group [Formula: see text], we prove that if, for some [Formula: see text], [Formula: see text] for all [Formula: see text], then the order of the nilpotent residual [Formula: see text] is bounded in terms of [Formula: see text].

BOUNDING THE ORDER OF THE NILPOTENT RESIDUAL OF A FINITE GROUP

The last term of the lower central series of a finite group $G$ is called the nilpotent residual. It is usually denoted by $\unicode[STIX]{x1D6FE}_{\infty }(G)$. The lower Fitting series of $G$ is defined by $D_{0}(G)=G$ and $D_{i+1}(G)=\unicode[STIX]{x1D6FE}_{\infty }(D_{i}(G))$ for $i=0,1,2,\ldots \,$. These subgroups are generated by so-called coprime commutators $\unicode[STIX]{x1D6FE}_{k}^{\ast }$ and $\unicode[STIX]{x1D6FF}_{k}^{\ast }$ in elements of $G$. More precisely, the set of coprime commutators $\unicode[STIX]{x1D6FE}_{k}^{\ast }$ generates $\unicode[STIX]{x1D6FE}_{\infty }(G)$ whenever $k\geq 2$ while the set $\unicode[STIX]{x1D6FF}_{k}^{\ast }$ generates $D_{k}(G)$ for $k\geq 0$. The main result of this article is the following theorem: let $m$ be a positive integer and $G$ a finite group. Let $X\subset G$ be either the set of all $\unicode[STIX]{x1D6FE}_{k}^{\ast }$-commutators for some fixed $k\geq 2$ or the set of all $\unicode[STIX]{x1D6FF}_{k}^{\ast }$-commutators for some fixed $k\geq 1$. Suppose that the size of $a^{X}$ is at most $m$ for any $a\in G$. Then the order of $\langle X\rangle$ is $(k,m)$-bounded.