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". The relation is written "H ser G" or "H is serial in G".
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".