Almost completely decomposable torsion free abelian groups

Proc. Amer. Math. Soc. 45(1974), pp. 41 - 47.

This was the first paper I ever wrote completely on my own. I started it during August, 1972, just after getting my degree at New Mexico State. I finished it sometime before the end of the fall semester 1972.

I was by no means sure that I could find a topic for research without any help. Always before, when I had tried to come up with ideas of my own I had wound up discovering things that were already part of the literature.

I had studied Butler's classic paper on what are now called ``Butler groups'' while taking Dave Arnold's course at New Mexico State. Since this seemed to be the one thing I knew something about, I wondered if somehow I could carry Butler's ideas a little further. I decided that it might be worth starting with the very simplest case of Butler groups, namely the almost completely decomposable ones. This seemed like a pretty simple-minded topic, but I thought that at the very least I might manage to learn something by investigating these extremely simple groups. Maybe I'd get an insight that would lead me in the direction of something more profound.

I managed to find some proofs that I really liked. The only trouble was, I realized, that I was essentially just repeating things that were already in Butler's paper. The big issue (as always) was trying to figure out what questions were open and might be worth addressing. I was led to the consideration of direct sum decompositions, and their well known non-uniqueness, because this was something I'd put a lot of effort into making sense of as part of Dave Arnold's course. I'd already noticed that almost all the classic pathological examples of direct sum decompositions were almost completely decomposable groups. Now I thought it would be worth trying to see if these direct sums were a little less unruly than they seemed.

In search of some kind of orderliness within almost completely decomposables, I started thinking about maximal completely decomposable subgroups. I was able to prove that the index in the whole group of a maximal completely decomposable subgroup was always the same, and I used the notation i(G) to denote this index. I then managed to prove a couple of theorems about i(G) which didn't seem earthshaking, but were not totally trivial. What would have been really worthwhile would have been to establish the converse of one of these theorems, which would have said that if you have two quasi-isomorphic almost completely decomposable groups G and H and if i(G)=i(H), then G and H in fact ressemble each other very closely. In language I would later introduce, they would be ``nearly isomorphic.'' Unfortunately, I couldn't prove this converse.

The Math Department at Kansas typed my paper up for me and made me a bunch of ugly purple ditto copies, which I mailed out to everyone I could think of who I might want to impress.

Unfortunately, about a week after I mailed out all those copies, I realized that my big theorem was false: the maximal completely decomposable subgroups do not in fact all have the same index in the overall group. And so I had to send out retractions to all the people who had got the paper.

This apparent disaster, however, was what actually made the paper successful. Because what I now realized I should do was to give those completely decomposable subgroups with minimal index a special name. I wracked my brain for a suitable word, considering and rejecting such terms as ``straightening subgroup'' and ``stabilizing subgroup.'' Finally, in despair, I settled on the phrase ``regulating subgroup,'' which seemed to me at the time to be utterly lame. As it turns out, though, nobody has ever suggested any alternative.

The fact that the paper described a new class of subgroups of almost completely decomposable groups and introduced a new word to describe these subgroups somehow made it much more interesting than the original fallacious version, even though it was in fact now weaker than the original version would have been if correct.

In any case, to make up for the theorem I had now lost, I managed to find a marvelous homological proof for the result that I hadn't been able to prove before, namely that if G and H are quasi-isomorphic almost completely decomposable groups and if i(G)=i(H), then G and H are nearly isomorphic. (Someone not familiar with the field will not appreciate this, possibly suggesting proverbs such as, ``A miss is as good as a mile,'' but it turns out that ``nearly isomorphic'' is a much stronger ressemblance than ``quasi-isomorphic.'')

At this point, I thought that I had proved just about everything one possibly could about almost completely decomposable groups, and therefore turned my attention in other directions. It turned out, though, that years later mathematicians such as Burkhardt, Mutzbauer, and Mader were able to use the idea of regulating subgroups to go far beyond my results, proving theorems that I had never even dreamed of. Obviously I would have preferred to have been smart enough to see all these further theorems myself in the first place. On the other hand, if I hadn't left a lot of openings for other mathematicians to improve on my results, then my paper would have been the end of the road for almost completely decomposable groups and Göbel probably would never have characterized it as ``historic.'' (I have to say, though, that if I had my choice of which of my papers would be characterized as ``historic,'' this one would be nowhere near the top of the list.)