Injectors in $$\pi $$-Separable Groups

Let $$\sigma = \{ {\sigma }_{i}: i \in I \}$$ σ = { σ i : i ∈ I } be a partition of the set $${\mathbb {P}}$$ P of all prime numbers. A subgroup X of a finite group G is called $$\sigma $$ σ -subnormal in G if there is a chain of subgroups $$\begin{aligned}X=X_{0} \le X_{1} \le \cdots \le X_{n}=G\end{aligned}$$ X = X 0 ≤ X 1 ≤ ⋯ ≤ X n = G where for every $$j=1, \dots , n$$ j = 1 , ⋯ , n the subgroup $$X_{j-1}$$ X j - 1 is normal in $$X_{j}$$ X j or $$X_{j}/{{\,\textrm{Core}\,}}_{X_{j}}(X_{j-1})$$ X j / Core X j ( X j - 1 ) is a $${\sigma }_{i}$$ σ i -group for some $$i \in I$$ i ∈ I . A group G is said to be $$\sigma $$ σ -soluble if every chief factor of G is a $${\sigma }_{i}$$ σ i -group for some $$i \in I$$ i ∈ I . The aim of this paper is to study two classes of finite groups based on the transitivity of the $$\sigma $$ σ -subnormality. Some classical and known results are direct consequences of our study.

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On two classes of generalised finite T-groups

Ballester‐Bolinches

Pedraza-Aguilera

Pérez-Calabuig

2023

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.

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Injectors in $$\pi $$-Separable Groups

Abstract: Let $$\pi $$ π be a set of primes. We show that $$\pi $$ π -separable groups have a conjugacy class of $${\mathfrak {F}}$$ F -injectors for suitable Fitting classes $${\mathfrak {F}}$$ F , which coincide with the usual ones when specializing to soluble groups.

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On two classes of generalised finite T-groups

On two classes of generalised finite T-groups

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