Normal series

2020 Mathematics Subject Classification: Primary: 20-XX [MSN][ZBL]

A normal series of a group $G$ is a series of normal subgroups $$G=H_1\supseteq H_2\supseteq \cdots \supseteq H_{n+1} = \{1\} $$ of $G$ (see Subgroup series). If each term of the series is normal not in the whole group but only in the preceding term, then the series is called subnormal. Apart from finite series one also considers infinite descending or ascending normal and subnormal series, the terms of which are indexed by (transfinite) ordinal numbers. One considers also more general normal and subnormal systems, the terms of which are indexed by the elements of an ordered set.

A factor of the series is the quotient group of some term of the series by the following (or preceding, if the series has ascending order of terms). The length of a series is the number of its factors other than the trivial one. A normal series that cannot be refined further is called a chief series (cf. Principal series) and a subnormal one a composition series (cf. Composition sequence). The factors of such series are called chief and composition factors. Two normal (subnormal) series are called isomorphic if a one-to-one correspondence can be set up between their factors such that corresponding factors are isomorphic. Any two normal (subnormal) series have isomorphic refinements (Schreier's theorem). In particular, any two chief (composition) series are isomorphic (the Jordan–Hölder theorem).

There is also another (older) terminology, in which a normal series is what is called above subnormal and for the concept called here a "normal series" one uses the term invariant series .

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