A Jordan-Hölder decomposition of a group $G$ is a filtration $$ G = G_1 \supset G_2 \supset \cdots \supset G_n = \{1\} $$ such that $G_{i+1}$ is a normal subgroup of $G_i$ and the quotient $G_i/G_{i+1}$ is a simple group for each $i$
