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- W4387432679 abstract "Let $sigma={sigma_j,:, jin J}$ be a partition of the set $mathbb{P}$ of all prime numbers. A subgroup $X$ of a finite group $G$ is~textit{$sigma$-subnormal} in $G$ if there exists a chain of subgroups $$X=X_0leq X_1leqldotsleq X_n=G$$ such that, for each $1leq ileq n-1$, $X_{i-1}trianglelefteq X_i$ or $X_i/(X_{i-1})_{X_i}$ is a $sigma_{j_i}$-group for some $j_iin J$. Skiba~[12] studied the main properties of $sigma$-subnormal subgroups in finite groups and showed that the set of all $sigma$-subnormal subgroups plays a relevant role in the structure of a finite soluble group. In [5], we laid the foundation of a general theory of $sigma$-subnormal subgroups (and $sigma$-series) in locally finite groups. It turns out that the main difference between the finite and the locally finite case concerns the behaviour of the join of $sigma$-subnormal subgroups: in finite groups, $sigma$-subnormal subgroups form a sublattice of the lattice of all subgroups [3], but this is no longer true for arbitrary locally finite groups. This is similar to what happens with subnormal subgroups, so it makes sense to study the class $mathfrak{S}_sigma^infty$ (resp. $mathfrak{S}_sigma$) of locally finite groups in which the join of (resp. of finitely many) $sigma$-subnormal subgroups is $sigma$-subnormal. Our aim is to study how much one can extend a group in one of these classes before going outside the same class (see for example Theorems~3.6, 3.8, 5.5 and 5.7). Also, $sigma$-subnormality criteria for the join of $sigma$-subnormal subgroups are obtained: similarly to a celebrated theorem of Williams (see [15]), we give a necessary and sufficient conditions for a join of two $sigma$-subnormal subgroups to always be $sigma$-subnormal; consequently, we show that the join of two orthogonal $sigma$-subnormal subgroups is $sigma$-subnormal (extending a result of Roseblade [11])." @default.
- W4387432679 created "2023-10-09" @default.
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- W4387432679 date "2023-10-05" @default.
- W4387432679 modified "2023-10-09" @default.
- W4387432679 title "Joins of $sigma$-subnormal subgroups" @default.
- W4387432679 doi "https://doi.org/10.48550/arxiv.2310.03391" @default.
- W4387432679 hasPublicationYear "2023" @default.
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