# Central automorphisms of groups act transitively on Krull-Schmidt decompositions

(Cross posted from math.SE)

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 $G=H_1\times \cdots \times H_n$. 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. A normal automorphism is also 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. They have several other characterizations as well. These I've rarely seen mentioned in texts, but they are simple and common enough.

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. His thesis is actually available online, and I can confirm that these results are in there (though mostly as an excuse to say that Wedderburn did a poor job proving them). 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.