Serial subgroup

In the mathematical field of group theory, a subgroup ''H'' of a given group ''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 of ''Y''.<ref name=Giovanni2001>{{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}}</ref> The relation is written ''H ser G'' or ''H is serial in G''.<ref name=Hartley1972>{{cite journal|last1=Hartley|first1=B.|title=Serial subgroups of locally finite groups|journal=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 }}</ref>

If the chain is finite between ''H'' and ''G'', then ''H'' is a 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 of ''G''; if descending, then ''H'' is a descendant subgroup of ''G''. If ''G'' is a locally finite group, then the set of all serial subgroups of ''G'' form a complete sublattice in the lattice of all normal subgroups of ''G''.<ref name=Hartley1972 />

==References==

Category:Subgroup properties