Serial subgroup

Let $H$ be a subgroup of a group $G$. A series of subgroups between $H$ and $G$, or, more briefly, a series between $H$ and $G$, is a set of subgroups of $G$,

$$S = \{ {A _ \sigma , B _ \sigma } : {\sigma \in \Sigma } \} ,$$

where $\Sigma$ is a linearly ordered set, such that

i) $H \subset A _ \sigma$, $H \subset B _ \sigma$ for all $\sigma \in \Sigma$;

ii) $G \setminus H = \cup _ \sigma ( B _ \sigma \setminus A _ \sigma )$;

iii) $A _ \sigma$ is a normal subgroup of $B _ \sigma$;

iv) $B _ \tau$ is a subgroup of $A _ \sigma$ if $\tau < \sigma$.

It follows that for all $\tau < \sigma$,

$$A _ \tau riangle\left B _ \tau \subset A _ \sigma riangle\left B _ \sigma \right .$$

and

$$B _ \sigma = \cap _ {\tau > \sigma } A _ \tau ,\ A _ \sigma = \cup _ {\tau < \sigma } B _ \tau ,$$

and for a finite series, indexed by $\{ 0 \dots n \}$, hence

$$B _ {i} = A _ {i+} 1 ,\ i = 0 \dots n- 1.$$

A subgroup $H$ is called serial if there is a series of subgroups between $H$ and $G$. If $G$ is finite, a subgroup $H$ is serial if and only if it is a subnormal subgroup. A subgroup $H$ is called an ascendant subgroup in $G$ if there is an ascending series of subgroups between $H$ and $G$, that is, a series whose index set $\Sigma$ is well-ordered.

References

[a1]	D.J.S. Robinson, "Finiteness condition and generalized soluble groups" , 1 , Springer (1972) pp. Chapt. 1