I'm looking for a modern reference to the subject line, preferably one that doesn't use Ore's generalizations to modular lattices.

To clarify terminology...

Suppose we have a group $G$ satisfying the descending and ascending chain conditions on normal subgroups. Then we can find indecomposable subgroups $H_1,...H_n$ of $G$ such that $G=H_1\times \cdots H_n$. This is often known as the Krull-Schmidt theorem, though Wedderburn, Remak, and Ore are also commonly associated to it. Moreover, $n$ is the same for any such decomposition, and terms in any two such decompositions are pairwise isomorphic and replaceable. This is found in most (graduate) texts for an introduction to group theory.

The proof is accompanied by Fitting's lemma and the concept of normal endomorphisms: endomorphisms commuting with the conjugation action (alternatively, composition commuting with all inner automorphisms). A normal automorphism is called a central isomorphism. They can all be expressed as the convolution of a morphism $G\to Z(G)$ with the identity on $G$, hence the name. This I've rarely seen mentioned in texts, but it is common enough in research to not warrant a particular reference itself.

However, the subject line is a fact I've never seen mentioned even as an exercise in modern texts: given any two decompositions there is a central automorphism taking one to the other. This is not terribly difficult to work out by playing with the Krull-Schmidt theorem and its proof, but I would like a modern reference if possible.

Wikipedia only mentions it was proved in Remak's thesis from 1911. The closest I have to a modern reference is Hall's book on group theory, but this has several shortcomings. It is over 40 years old; he uses Ore's modular lattice proof (which won't quite work for what I want); and he only states that individual terms in two decompositions are pairwise centrally isomorphic, which is not as strong a statement as the central automorphisms acting transitively on the decompositions.

Does a reference better suiting my needs exist? I'd very much like one. Even a book that leaves these to the exercises would be better than what I currently have.

  • $\begingroup$ Could you elaborate on the characterization you mention for normal endomorphisms as convolutions of central endomorphisms with the identity? Where can I find this? $\endgroup$
    – Arrow
    Dec 26, 2016 at 0:01
  • $\begingroup$ @Arrow Given any $\phi\in\textrm{Hom}(G,Z(G))$, consider the map $g\mapsto \phi(g)g^{-1}$ and show it is an endomorphism of $G$. It is also a bijection if and only if $\phi$ has no fixed points, which is guaranteed if $G$ has no abelian direct factors (by Fitting's lemma). Conversely, you can show by a direct check that given any central automorphism $\psi$ of $G$ then $g\mapsto \psi(g)g$ is an element of $\textrm{Hom}(G,Z(G))$. These two maps are inverses of each other precisely when $G$ has no abelian direct factors. As for a reference, the only one that comes to mind is Adney & Yen. $\endgroup$ Dec 26, 2016 at 1:36


You must log in to answer this question.

Browse other questions tagged .