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subnormal series

A chain $\{e\}=H_0 \trianglelefteq H_1 \trianglelefteq \cdots \trianglelefteq H_n = G$ where each $H_i$ is normal in $H_{i+1}$ specifically -- not necessarily normal in $G$ itself.

Examples

$\{e\} \trianglelefteq A_3 \trianglelefteq S_3$ .