Difference between revisions of "Subnormal subgroup"
From Encyclopedia of Mathematics
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''attainable subgroup'' | ''attainable subgroup'' | ||
| − | Any member of any [[ | + | 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==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> M.I. Kargapolov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[1]</TD> <TD valign="top"> M.I. Kargapolov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian)</TD></TR> | ||
| + | </table> | ||
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A subnormal subgroup is also called a subinvariant subgroup. | A subnormal subgroup is also called a subinvariant subgroup. | ||
| − | A subnormal subgroup of | + | 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. [[#References|[a1]]]. |
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Suzuki, "Group theory" , '''1–2''' , Springer (1986)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J.C. Lennox, S.E. Stonehewer, "Subnormal subgroups of groups" , Clarendon Press (1987)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> D.J.S. Robinson, "A course in the theory of groups" , Springer (1982)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Suzuki, "Group theory" , '''1–2''' , Springer (1986)</TD></TR> | ||
| + | <TR><TD valign="top">[a2]</TD> <TD valign="top"> J.C. Lennox, S.E. Stonehewer, "Subnormal subgroups of groups" , Clarendon Press (1987)</TD></TR> | ||
| + | <TR><TD valign="top">[a3]</TD> <TD valign="top"> D.J.S. Robinson, "A course in the theory of groups" , Springer (1982)</TD></TR> | ||
| + | </table> | ||
| + | |||
| + | {{TEX|done}} | ||
Revision as of 19:37, 28 February 2018
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://encyclopediaofmath.org/index.php?title=Subnormal_subgroup&oldid=15071
Subnormal subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Subnormal_subgroup&oldid=15071
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