# Subnormal subgroup

From Encyclopedia of Mathematics

*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) |

**How to Cite This Entry:**

Subnormal subgroup.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Subnormal_subgroup&oldid=42877

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