Serial subgroup
From HandWiki
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.[1] The relation is written H ser G or H is serial in G.[2]
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.[2]
See also
- Characteristic subgroup
- Normal closure
- Normal core
References
- ↑ de Giovanni, F.; A. Russo; G. Vincenzi (2002). "GROUPS WITH RESTRICTED CONJUGACY CLASSES". Serdica Math. J. 28: 241–254.
- ↑ 2.0 2.1 Hartley, B. (24 October 2008). "Serial subgroups of locally finite groups". Mathematical Proceedings of the Cambridge Philosophical Society 71 (2): 199–201. doi:10.1017/S0305004100050441. Bibcode: 1972PCPS...71..199H.
 |  |
