Fitting Theorem

Definition

Let $G$ be a group, and let $\Omega \subseteq \mathrm{End}(G)$ be a set of endomorphisms of $G$. Then we say $G$ is an $\Omega$-group.

Definition

For a $\Omega$-group $G$, we say $\mu \in \mathrm{End}(G)$ is an $\Omega$-endomorphism if $\mu \omega =\omega \mu$ for any $\omega \in \Omega$. In particular, we say $\mu$ is a normal endomorphism if $\Omega =\mathrm{Inn}(G)$ and $\mu$ is an $\Omega$-endomorphism.

Fitting theorem

Let $G$ be an $\Omega$-group, and let $\mu$ be an $\Omega$-endomorphism such that the chains $\{G^{{\mu }^k}{\}}_{k\in \mathbb{N}}$ and $\{\ker {\mu }^{\ell }{\}}_{\ell \in \mathbb{N}}$ stabilize. Then there exists $n\gg 0$ such that

$$G=G^{{\mu }^n}\times \ker {\mu }^n.$$

Example. Let $({\mathbb{C}}^n,+)$ be an abelian group, and let $\Omega =\mathbb{C}$, where $v\mapsto cv$ for any $c\in \mathbb{C}$.

• Any linear transformation $\mu$ is a $\mathbb{C}$-endomorphism. By Fitting theorem, there exists $n\in {\mathbb{N}}_{+}$ such that $V=\ker ({\mu }^n)\oplus \mathrm{im}({\mu }^n)$.
• If $\lambda$ is an eigenvalue of $\mu$, then $\beta =\lambda I-\mu$ is also a linear transformation and there exists $n^{\prime }$ such that $V=\ker ({\beta }^{n^{\prime }})\oplus \mathrm{im}({\beta }^{n^{\prime }})$. Remark that $\ker ({\beta }^{n^{\prime }})$ is the generalized eigenspace of $\lambda$. Repeat this procedure, and $V$ can be written as direct sum of generalized eigenspaces.
• Finally, we can decompose each generalized eigenspaces into several indecomposable subspaces. By ^w0xmgq, the decomposition is unique. Now we get the Jordan normal form of $\mu$.