Group Theory/Subnormal subgroups and series

{{definition|subnormal subgroup|Let $$G$$ be a group. A subgroup $$H \in G$$ is called subnormal subgroup if and only if there exists

{{definition|subnormal series|Let $$G$$ be a group. Then a subnormal series is a finite family of subgroups $$H_0, H_1, \ldots, H_n \le G$$ such that
 * $$\{e\} = H_0 \triangleleft H_1 \triangleleft \cdots \triangleleft H_n = G$$,

where $$e \in G$$ is the identity.}}

{{definition|composition series|Let $$G$$ be a group. A composition series of $$G$$ is a subnormal series
 * $$\{e\} = H_0 \triangleleft H_1 \triangleleft \cdots \triangleleft H_n = G$$

of $$G$$ such that for all $$k \in \{1, \ldots, n\}$$ the quotient group $$H_k / H_{k-1}$$ is simple.}}

{{theorem|Schreier refinement theorem|Let $$G$$ be a group, and let
 * $$\{e\} = H_0 \triangleleft H_1 \triangleleft \cdots \triangleleft H_n = G$$

be a subnormal series of $$G$$.}}

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