Group theory and generalizations Definition

Subnormal series

Builds on Normal subgroup Quotient group

Definition

Let $G$ be a group. A subnormal series of $G$ is a finite sequence of subgroups $\{e\} = G_0 \trianglelefteq G_1 \trianglelefteq \cdots \trianglelefteq G_n = G$ , where each $G_{i-1}$ is a normal subgroup of $G_i$ . The quotient groups $G_i / G_{i-1}$ are the factors of the series, and $n$ is its length.

Remarks

The subgroup $G_{i-1}$ needs to be normal only in $G_i$ , not in $G$ . When each $G_i$ is normal in $G$ , the series is called a normal series. For example, $\{e\} \trianglelefteq V \trianglelefteq A_4 \trianglelefteq S_4$ is a normal series of $S_4$ , where $V$ is the Klein four-group in $S_4$ .

Tag	`096HN851YZ`
URI	`m2://096HN851YZ`
Aliases	`subinvariant-series`
Kind	definition
Writing	W0 stub
MSC	20E15
Defines	Subnormal series
Uses	Normal subgroup, Quotient group

Subnormal series

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Remarks

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