Krull Schmidt theorem for groups Krull-Schmidt theorem says that if a group G has both chain conditions, and $G = H_1 \times H_2 \times \dots \times H_s = K_1 \times K_2 \times \dots \times K_t$ be two decompositions of G into indecomposable factors, then $s = t$, and there is a reindexing so that $H_i \cong K_i, \forall i$.
Moreover, given any $r$  between  1 and $s$, the reindexing maybe chosen so that $G = H_1 \times H_2 \times \dots H_r \times K_{r+1} \times K_s$.
Now, Rotman in his book makes a remark that "The last conclusion is stronger than saying that the factors are determined up to isomorphism; one can replace factors of one decomposition by suitable factors from the other."  I have trouble understanding what this means, and appreciating the subtlety of this statement. Could someone explain what it means, perhaps by an example as well?
 A: What Rotman means is that you can always substitute one of the K's for one of the H's (and keep doing that) and still have a direct product.  In other words, you can transform from one direct product representation to the other by changing just one factor at a time and as long as you choose carefully, all the intermediate choices will also be direct products.
For example, if $G=4^3$ (the direct product of 3 cyclic groups of order 4), and $G=H_1\times H_2\times H_3 = K_1\times K_2\times K_3$ all factors cyclic of order 4, then one of the $K_i$ must intersect $H_1\times H_2$ trivially so that $H_1\times H_2\times K_i$ is direct (and equals $G$).  That's the content of Rotman's "stronger than".
Counterexamples when there is no composition series include $\Bbb{Q}/\Bbb{Z}$ and the infinite direct product of $\Bbb{Z}$ with itself.  See Nonexample to Krull-Schmidt Groups 
Note also that the Krull-Schmidt theorem also works for groups with operators.  (See Isaacs algebra text, though I think there is a group theory text, maybe Hall or Magnus, that also has that result...and certainly Huppert Vol 1 has, if you read German)  The result for groups with operators is nice since it includes the result for modules.  (In all cases, a "composition series" is required.)
