Subnormal subgroup

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.

References

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

Comments

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].

References

[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)