Abstract Algebra: Theory and Applications, Remixed!

Given a group, we can look at subgroups. We say that a sequence of subgroups

is a subnormal series provided each $H_i$ is normal in $H_{i+1}\text{,}$ and a normal series if each $H_i$ is normal in $G\text{.}$

A non-trivial group $G$ is called simple provided it has no non-trivial normal subgroups. We say that a subnormal series is a composition series and that a normal series is a principle series if every quotient group $H_{i+1}/H_i$ is simple.