Serial subgroup

{{technical|date=October 2013}}

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.{{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|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 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.