Definition:Normal Series
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Definition
Let $G$ be a group whose identity is $e$.
A normal series for $G$ is a sequence of (normal) subgroups of $G$:
- $\left\{{e}\right\} = G_0 \triangleleft G_1 \triangleleft \cdots \triangleleft G_n = G$
where $G_{i-1} \triangleleft G_i$ denotes that $G_{i-1}$ is a proper normal subgroup of $G_i$.
Factors
The factor groups (or just factors) of a normal series:
- $\left\{{e}\right\} = G_0 \triangleleft G_1 \triangleleft \cdots \triangleleft G_n = G$
are the quotient groups:
- $G_0 / G_1, G_1 / G_2, \ldots, G_{i-1} / G_i, \ldots, G_{n-1} / G_n$
Length
The length of a normal series is the number of (normal) subgroups which make it.
Alternative Names
A normal series is also known as:
- A subnormal series
- A normal tower
- A subinvariant series
Make the distinction here between a series where all $G_k$ are normal subgroups of $G$ itself, and those where they do not have to be. The former case is often called a subnormal series, the latter a normal series. See Wikipedia's Subgroup Series for background.
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Sources
- Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 71$
