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The Krull—Schmidt theorem states that every group with ACC and DCC on its normal subgroups has only one decomposition into the product of directly indecomposable groups (in particular, this holds for finite groups).

I am looking for the examples of groups that have more than one direct product decomposition. The final goal is to give some of such examples as illustrations in a basic group theory course, so I particularly welcome examples with a simple description, e.g. transformations of some geometric configurations. Anyway, having a big list of examples is always useful.

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    $\begingroup$ Wikipedia claims that "There are $\mathbf{Z}$-submodules of $\mathbf{Q}^4$ which are simultaneously direct sums of two indecomposable modules and direct sums of three indecomposable modules," although I have been unable to find any reference or construction. ​ ​ $\endgroup$ – user57159 Oct 27 '16 at 13:06
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Bjarni Jonsson had a result somewhat like the one mentioned by Ricky Demer. It appears in

Jónsson, Bjarni On direct decompositions of torsion-free abelian groups. Math. Scand. 5 1957, 230-235.

I will copy the description of his group here.

Consider a four dimensional vector space $V$ over the field of rational numbers, let $\{x, y, z, u\}$ be a basis for $V$, and let \[x' = 3x-y\quad \textrm{and}\quad y' = 2x-y.\] Then $\{x', y', z, u\}$ is also a basis for $V$, and \[x = x'-y' \quad \textrm{and}\quad y = 2x'-3y'.\] Let $A, B, C, D$ consist of all those elements of $V$ which can be written in the forms \[\begin{array}{rl} \frac{a}{5^n}x, \quad &\frac{b}{5^n}y+\frac{c}{7^n}z+\frac{d}{11^n}u+ \frac{e}{3}(y+z)+\frac{f}{2}(y+u),\\ \frac{a}{5^n}x'+\frac{c}{7^n}z+\frac{e}{3}(x'-z), \quad &\frac{b}{5^n}y'+\frac{d}{11^n}u+\frac{f}{2}(y'-u), \end{array} \] respectively, with $a, b, c, d, e, f$ and $n$ being integers.

Jonsson proves that $A\oplus B = C\oplus D$, that $A, C, D$ are indecomposable (he states without proof that the same is true for $B$), and that $A$ is the only one of them that is embeddable in $\mathbb Q$.

$\ldots$ I particularly welcome examples with a simple description $\ldots$

Maybe Jonsson's example does not satisfy this criterion. If you only need it for illustration purposes, here is an easy example from a category of modules.

Let $R$ be an integral domain that has comaximal ideals $I$ and $J$ that are not principal. For example, you could take $R=\mathbb C[x,y]$, $I = (x,y)$, $J = (x-1,y)$. The exact sequence of $R$-modules

$$ 0\to I\cap J\to I\oplus J\stackrel{\alpha}{\to} R\to 0, \quad\alpha(i,j)=i+j, $$

splits because the right term is free. Hence $I\oplus J\cong R\oplus (I\cap J)$. Each of $R, I, J, I\cap J$ is indecomposable, since they are ideals in a domain. $R$ is cyclic, but $I$ and $J$ are not, so at least one term on the left is not isomorphic to at least one term on the right.

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