Jordan-Hölder decomposition

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 $i$ .

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