Subgroup series

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

A subgroup series of a group $G$ is a finite chain of subgroups of $G$ contained in each other: $$E=G_0\subseteq G_1\subseteq \cdots \subseteq G_n=G\label{1}$$

or

$$G=G_1\supseteq G_2\supseteq \cdots \supseteq G_{n+1}=E.$$ One also considers infinite chains of imbedded subgroups (increasing and decreasing), which may be indexed by a sequence of numbers or even by elements of an ordered set. They are often called subgroup systems (cf. Subgroup system).

An important part in group theory is played by subnormal, normal and central series. A subgroup series (1) is called a subnormal series if each group in the chain is a normal subgroup of the subsequent term. If also each subgroup $G_i$, $i=0,\dots,n$, is normal in $G$, the series (1) is called a normal series in $G$. There is also a different terminology, in which the name normal series is given to what is here called subnormal, while for the second concept defined here one uses the term "invariant subgroup seriesinvariant series" . The quotient groups $G_{i+1}/G_i$ are called factors, while the number $n$ is the length of the subnormal series. A normal series (1) is called a central series if all its factors are central, i.e. $G_{i+1}/G_i$ lies in the centre of the group $G/G_i$ for all $i$, or, which is equivalent, the commutator of $G_{i+1}$ and $G$ lies in $G_i$ for all $i$. If $G_{i+1}/G_i$ is the centre of the group $G/G_{i}$ (respectively, if the commutator of $G_{i+1}$ and $G$ coincides with $G_i$) for all $i$, then the series (1) is called the upper central series (respectively, the lower central series) of $G$. Let a subnormal (respectively, normal or central) series be given in a group $G$ together with a certain subgroup $H\subseteq G$, and let $H_i=G_i\cap H$, $i=0,\dots,n$. Then the chain

$$E=H_0\subseteq H_1\subseteq \cdots \subseteq H_n=H$$ is a subnormal (respectively, normal or central) series in $H$, while the factors of this series are isomorphic to subgroups of the corresponding factors in the series (1). If $G/N$ is a quotient group of $G$, the chain

$$E=G_0N/N\subseteq G_1N/N\subseteq \cdots \subseteq G_nN/N=G/N$$ is a subnormal (respectively, normal or central) series in $G/N$, and the factors in this series are homomorphic images of the corresponding factors in the series (1).

Two subnormal (in particular, normal) series in a group are said to be isomorphic if they have the same length and if there is a bijection between their factors such that corresponding factors are isomorphic. If every subgroup of one of the series coincides with one of the subgroups of the other, the second series is called a refinement of the first. A normal series which cannot be refined is called a principal series (or chief series); while a subnormal series which cannot be refined is called a composition series. The factors in these series are called chief and composition factors, respectively. Any two subnormal (respectively, normal or central) series of a group have isomorphic subnormal (respectively, normal or central) refinements. In particular, any two chief (composition) series are isomorphic (see Jordan–Hölder theorem).

References

[Ch]	S.N. Chernikov, "Groups with given properties of subgroup systems", Moscow (1980) (In Russian)
[KaMe]	M.I. Kargapolov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups", Springer (1979) (Translated from Russian) MR0551207 Zbl 0549.20001
[Ro]	D.J.S. Robinson, "A course in the theory of groups", Springer (1980) MR1357169 MR1261639 MR0648604 Zbl 0836.20001 Zbl 0483.20001
[Su]	M. Suzuki, "Group theory", 1, Springer (1986) MR0815926 Zbl 0586.20001