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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
editReferences
edit- ^ de Giovanni, F.; A. Russo; G. Vincenzi (2002). "GROUPS WITH RESTRICTED CONJUGACY CLASSES". Serdica Math. J. 28: 241–254.
- ^ a b Hartley, B. (24 October 2008) [1972]. "Serial subgroups of locally finite groups". Mathematical Proceedings of the Cambridge Philosophical Society. 71 (2): 199–201. Bibcode:1972PCPS...71..199H. doi:10.1017/S0305004100050441. S2CID 120958627.
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