Krull Schmidt, ACC and DCC

ACC and DCC

Throughout this topic, a "chain" consists of normal subgroups of G, linearly ordered by containment. This is different from the normal series that we saw earlier, where each subgroup must be normal in the subgroup containing it. The subgroups of our chain need only be normal in G.

If G has no infinite ascending chain it has the "ascending chain condition", also known as acc. If G has no infinite descending chain it has the "descending chain condition", also known as dcc. This is consistent with the corresponding definitions for modules.

Note that G could be acc and dcc, and still have infinite chains of subgroups that are not normal. Consider the alternating group on infinitely many letters. This group is simple, hence it is acc and dcc. The subgroups A_n, as n runs from 1 to infinity, form an ascending chain, and the subgroups that exclude the first n letters form a descending chain.

Let Q be a quotient group of G. Since normal subgroups of Q pull back to normal subgroups of G, an infinite chain in Q lifts to an infinite chain in G. Therefore Q inherits acc or dcc from G.

Direct Product

Let G = K*Q, a special case of kernel and quotient. By the above, Q and K inherit acc or dcc from G. The converse is also true. If K and Q are acc or dcc then so is G. See the corresponding proof for modules.

This generalizes to a finite direct product of groups. The product is acc or dcc iff the same holds for each component.

Let J be a normal subgroup of K. Conjugate x*J/x, and the K component stays in J, while the Q komponent drops out. Thus J is normal in G. A chain of normal subgroups in K becomes, without modification, a chain in G.