### Lecture 8: Introduction.

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 --- header-includes: | \makeatletter \def\input@path{{source/}} \input{latex-headers.tex} --- \newpage # LECTURE 8 # Solvability \sectionmark{\textit{2019.04.12}} \subsectionmark{\textit{Lecture 8}} In this lecture, we will concern ourselves with the question of which groups are solvable. First we recall what it means to be solvable. **Definition 1.** (Solvable) A finite group $G$ is **solvable** if there exist subgroups of $G$, $S_0, \ldots, S_n$ such that \begin{align*} \{\mathrm{id}\} = S_0 \triangleleft S_1 \triangleleft \ldots \triangleleft S_{n-1} \triangleleft S_n = G \end{align*} such that every quotient $S_{j+1}/S_j$ is abelian. *Remark.* Without the requirement of the quotient being abelian, we call such a sequence of normal subgroups a **cascading tower of normal subgroups** or a **subnormal series**. Sometimes however, it is easier to work with an equivalent characterisation. **Definition 2.** (Solvable) A finite group $G$ is **solvable** if there exists a cascading tower of normal subgroups in $G$ with $S_0 = \{\mathrm{id}\}$ and $S_n = G$ such that every quotient $S_{j+1}/S_j$ is cyclic. *Remark.* In both definitions, the quotient is well defined, as the subgroups are normal to their neighbours. *Remark.* (Normality is not transitive) In a cascading tower of normal subgroups we only require $S_j$ to be normal in $S_{j+1}$ (for all $j$). This however does **not** mean that for any \$i
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