# Serial subgroup
> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Serial_subgroup
> Markdown URL: https://mediated.wiki/source/Serial_subgroup.md
> Source: https://en.wikipedia.org/wiki/Serial_subgroup
> Source revision: 1277631402
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)
{{technical|date=October 2013}}
In the [mathematical](/source/mathematics) field of [group theory](/source/group_theory), a [subgroup](/source/subgroup) ''H'' of a given [group](/source/group_(mathematics)) ''G'' is a '''serial subgroup''' of ''G'' if there is a chain ''C'' of subgroups of ''G'' extending from ''H'' to ''G'' such that for consecutive subgroups ''X'' and ''Y'' in ''C'', ''X'' is a [normal subgroup](/source/normal_subgroup) of ''Y''.[{{cite journal|last1=de Giovanni|first1=F.| last2=Russo | first2=A. |last3=Vincenzi |first3=G. |title=Groups with restricted conjugacy classes|journal=Serdica Mathematical Journal|year=2002|volume=28|issue=3|pages=241–254|url=http://www.math.bas.bg/serdica/2002/2002-241-254.pdf}}] The relation is written ''H ser G'' or ''H is serial in G''.[{{cite journal|last1=Hartley|first1=B.|title=Serial subgroups of locally finite groups|journal=[Mathematical Proceedings of the Cambridge Philosophical Society](/source/Mathematical_Proceedings_of_the_Cambridge_Philosophical_Society)|date=March 1972|volume=71|issue=2|pages=199–201|doi=10.1017/S0305004100050441|bibcode=1972PCPS...71..199H|s2cid=120958627 }}]
If the chain is finite between ''H'' and ''G'', then ''H'' is a [subnormal subgroup](/source/subnormal_subgroup) of ''G''. Then every subnormal subgroup of ''G'' is serial. If the chain ''C'' is well-ordered and ascending, then ''H'' is an [ascendant subgroup](/source/ascendant_subgroup) of ''G''; if descending, then ''H'' is a [descendant subgroup](/source/descendant_subgroup) of ''G''. If ''G'' is a [locally finite group](/source/locally_finite_group), then the set of all serial subgroups of ''G'' form a [complete sublattice](/source/complete_lattice) in the [lattice](/source/lattice_(order)) of all normal subgroups of ''G''.