Direct Decomposition of Groups

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.

• 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. – Pedro Sánchez Terraf Jan 10 '17 at 0:59
• Ah - thanks very much! – Mike Jan 10 '17 at 9:25
• You're welcome. If this is enough of an answer for you, I might post it as such so you can close this thread. – Pedro Sánchez Terraf Jan 10 '17 at 18:59
• That would be nice, thanks again – Mike Jan 12 '17 at 13:03

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.
• Sorry about that, I have forgotten to add the condition that the lattice of congruences on $G = G_1 \times G_2 \times \cdots \times G_n$ has finite length. – Mike Jan 9 '17 at 23:38