# 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 SectionOfAGroup 2013-03-22 17:15:04 2013-03-22 17:15:04 yark (2760) yark (2760) 11 yark (2760) Definition msc 20F99 section quotient of a subgroup involved in