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Given a group G

  1. How can one tell if it could have been formed as the direct product of sub groups.
  2. If so what are the groups.
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    $\begingroup$ How are you given this group? The question is much too vague, voting to close. $\endgroup$
    – Igor Rivin
    Jun 21 '11 at 4:02
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    $\begingroup$ In general, there will be a system of congruences of any algebraic structure which correspond to a direct product representation if there is a nontrivial one. In general, factorization of algebraic structures is non-unique and not easily determined; groups are a tad easier, but one must scrutinize the congruence (normal subgroup) lattice. McKenzie, McNulty and Taylor cover quite a bit of ground in this area in Chapter 5 of their book "Algebras, Lattices, Varieties". Gerhard "Email Me About System Design" Paseman, 2011.06.20 $\endgroup$ Jun 21 '11 at 4:36
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Here is a `simple' procedure.

  1. Enumerate all non-trivial normal subgroups, $N$, of $G$.

  2. Check if the extension $N \stackrel{f}{\to} G \stackrel{h}{\to} G/N$ is split; ie there exists some homomorphism $s : G/N \to G$ such that $s h = id_{G/N}$.

  3. Finally, show that the image of $s$ is a normal subgroup of $G$.

The hard part of this whole process is step 2, which basically requires you to figure out some way of picking a representative element out of each of the cosets of $G/N$. If the extension is indeed split, there is really only one way to do this (ie map each element $x \in G/N$ to the `pair' $(x, e)$ where $e$ is the identity of $N$); and if it is not split then it is impossible.

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