Direct Decomposition of Groups

Question

A group $G$ is called directly irreducible if $G \simeq A \times B$ implies $G \simeq A$ or $G \simeq B$. I am looking for proof of the following theorem:

• If $G = G_1 \times G_2 \times \cdots \times G_n \simeq H_1 \times H_2 \times \cdots \times H_m$ , where the $G$'s and $H$'s are directly irreducible groups, and the lattice of congruences on $G$ has finite-length (it satisfies both ACC and DCC), then $n = m$ and there exists a permutation $\sigma \in S_n$ such that $G_i \simeq H_{\sigma(i)}$.

A proof, or a reference to a proof, would be much appreciated.

I have been told that there is a proof that involves the Kurosh-Ore Theorem, on modular lattices. This is specifically the proof that I am looking for, but anything will do, thanks.

Answer 1

The mandatory reference is Algebras, Lattices, Varieties (vol I), by McKenzie, McNulty and Taylor. In Section 2.3 they start with the subject, and Kurosh-Ore is Theorem 2.33. All of Chapter 5 is devoted to unique factorization, and you might be interested in Section 5.3.

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(This was a comment to the question that apparently is satisfactory answer to the OP, so I'm reposting as such.)

Answer 2

This is just now true. There exist a group $G$ and "directly irreducible" groups $A, B, C, D$ such that $G$ is isomorphic to both $A \times B$ and $C \times D$, these being non-isomorphic direct decompositions. See for example $\S$ 42 in A.G. Kurosh, The theory of groups, Volume two, second english edition, translated by K.A. Hirsch, AMS Chelsea Publishing, 1960.

If you are ready to excuse my French, see also this and other examples in Y. Cornulier and P. de la Harpe, Décompositions de groupes par produit direct et groupes de Coxeter, in Geometric group theory, 75-102, Trends in Math., Birkhäuser, Basel, 2007.