Jordan-Hölder decomposition theorem

Every finite group $G$ has a filtration

$$G\supset G_0\supset \mathrm{\cdots }\supset G_n=\{1\},$$

where each $G_{i+1}$ is normal in $G_i$ and each quotient group $G_i/G_{i+1}$ is a simple group. Any two such decompositions of $G$ have the same multiset of simple groups $G_i/G_{i+1}$ up to ordering.

A filtration of $G$ satisfying the properties above is called a Jordan–Hölder decomposition of $G$.