Subnormal subgroup

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attainable subgroup

Any member of any subnormal series of a group. To indicate the subnormality of a subgroup $H$ in a group $G$, the notation $H \lhd\!\lhd G$ is used.


[1] M.I. Kargapolov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian)


A subnormal subgroup is also called a subinvariant subgroup.

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. [a1].


[a1] M. Suzuki, "Group theory" , 1–2 , Springer (1986)
[a2] J.C. Lennox, S.E. Stonehewer, "Subnormal subgroups of groups" , Clarendon Press (1987)
[a3] D.J.S. Robinson, "A course in the theory of groups" , Springer (1982)
How to Cite This Entry:
Subnormal subgroup. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by N.N. Vil'yams (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article