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 $,

$$ A _ \tau \lhd B _ \tau \subset A _ \sigma \lhd B _ \sigma $$

and

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

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

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

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 conditions and generalized soluble groups. Part 1". Ergebnisse der Mathematik und ihrer Grenzgebiete. Band 62. Springer (1972) Zbl 0243.20032 Chap. 1