Subnormal series

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2020 Mathematics Subject Classification: Primary: 20D30 Secondary: 20D35 [MSN][ZBL]

A subnormal series (or subinvariant series) of a group $G$ is a subgroup series $$ E = G_0 \le G_1 \le \cdots \le G_n = G $$ in which each subgroup $G_i$ is a normal subgroup of $G_{i+1}$. The quotient groups $G_{i+1}/G_i$ are called factors, and the number $n$ is called the length of the subnormal series. Infinite subnormal series have also been studied (see Subgroup system).

A subnormal series that cannot be refined further is called a composition series, and its factors are called composition factors. Any two subnormal series of a group have isomorphic refinements and in particular, any two composition series are isomorphic (see Jordan–Hölder theorem).

A subnormal subgroup (also subinvariant, attainable or accessible) of $G$ is a subgroup that appears in some subnormal series of $G$. To indicate the subnormality of a subgroup $H$ in a group $G$, the notation $H \lhd\!\lhd G$ is used.

The property of a subgroup to be subnormal is transitive. An intersection of subnormal subgroups is again a subnormal subgroup. The subgroup generated by two subnormal subgroups need not be subnormal. A group $G$ all subgroups of which are subnormal satisfies the normalizer condition, i.e. all subgroups differ from their normalizers (cf. Normalizer of a subset). Such a group is therefore locally nilpotent.

A subnormal subgroup of $G$ that coincides with its commutator subgroup and whose quotient by its centre is simple is called a component of $G$. The product of all components of $G$ is known as the layer of $G$. It is an important characteristic subgroup of $G$ in the theory of finite simple groups, see e.g. [6].


[1] M.I. Kargapolov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian)
[2] A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian)
[3] M. Hall jr., "The theory of groups" , Macmillan (1959) pp. Sect. 8.4
[4] J.C. Lennox, S.E. Stonehewer, "Subnormal subgroups of groups" , Clarendon Press (1987)
[5] D.J.S. Robinson, "A course in the theory of groups" , Springer (1982)
[6] M. Suzuki, "Group theory" , 1–2 , Springer (1986)
How to Cite This Entry:
Attainable subgroup. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.M. Kopytov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article