serial subgroup

《Lecture Notes on Bayesian Estimation and Classification》 · 第 377—378 页 · Section 12.4「Classes of Groups Defined by General Series」，Definition 及小节「Composition Series」「Serial Subgroups」

Let $G$ be an $\Omega$-operator group. By a (general) $\Omega$-series of $G$ we shall mean a set $\mathbf S$ of subgroups, called the terms of $\mathbf S$, which is linearly ordered by inclusion and which satisfies the following conditions: (i) If $1\neq x\in G$, there are terms of $\mathbf S$ which do not contain $x$ and the union of all such terms is a term $V_x$ of $\mathbf S$. (ii) If $1\neq x\in G$, there are terms of $\mathbf S$ which contain $x$ and the intersection of all such terms is a term $\Lambda_x$ of $\mathbf S$. (iii) $V_x\triangleleft\Lambda_x$. (iv) Each term of $\mathbf S$ is of the form $V_x$ or $\Lambda_x$ for some $x\neq1$ in $G$. […] When $\Omega$ is empty, we speak of a series and a composition series: when $\Omega$ is the group of inner automorphisms, we speak of a normal series and a principal series. […] A subgroup which occurs in some series of a group $G$ is called serial.