section of a group

A section of a group $G$ is a quotient (http://planetmath.org/QuotientGroup) of a subgroup of $G$. That is, a section of $G$ is a group of the form $H/N$, where $H$ is a subgroup of $G$, and $N$ is a normal subgroup of $H$.

A group $G$ is said to be involved in a group $K$ if $G$ is isomorphic to a section of $K$.

The relation ‘is involved in’ is transitive (http://planetmath.org/Transitive3), that is, if $G$ is involved in $K$ and $K$ is involved in $L$, then $G$ is involved in $L$.

Intuitively, ‘$G$ is involved in $K$’ means that all of the structure of $G$ can be found inside $K$.

Title	section of a group
Canonical name	SectionOfAGroup
Date of creation	2013-03-22 17:15:04
Last modified on	2013-03-22 17:15:04
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	11
Author	yark (2760)
Entry type	Definition
Classification	msc 20F99
Synonym	section
Synonym	quotient of a subgroup
Defines	involved in