UPDATED SUBSTANTIALLY 7/6/2023
[Original Version Posted 1/22/2019]

PDF Notes: Semidirect Products, Group Extensions, Split Exact Sequences, and all that

If you suffer physics-brain and don’t know anything about semidirect products or group extensions, but get the sense direct products just aren’t cutting it — these are the notes for you.

Update: On reexamination, my original post on the subject was too brief and simplistic. Among other things, it neglected to explicitly construct the multiplication on a group extension. In the process of doing so, I identified numerous other shortcomings and omissions. As a result, I’ve replaced it with a far more thorough discussion. Unfortunately, at 50 pages it has expanded beyond the point where the primitive wordpress support for latex suffices. The notes now are provided as a pdf (linked at the top and bottom of this post). This has the added benefit of allowing me to the color-code proofs and comments and examples, allowing easier reading of the key elements.

The purpose of this post is to cure one aspect of physics-brain. In a typical physics education, we learn a sloppy version of linear algebra, basic group theory, and (certain) differential equations many times — and little else math-related. As a result, we develop numerous bad habits and suffer from a complete ignorance of certain important areas of mathematics. If two groups are being combined, it’s via a direct product. After all, what else is there? Unfortunately, this only works until it doesn’t — which is pretty much anywhere it matters. For example, even O(3,1) is not a direct product of the 2 copies of Z2 and SO+(3,1). You’ve probably heard of O(3,1). It matters.

Many of the things we naively assume would be direct products actually are semidirect products or general group extensions, and the result can be a lot of confusion when things don’t work as expected. Besides the construction of basic physical groups such as the Poincare group, this also arises in the study of quantum mechanics. Part of the reason we can pretend to work in Hilbert space when the actual state space is a projective Hilbert space is that the projective representations of a group lift to unitary representations of a different group. The latter is a particular group extension (in fact, a central extension) of the original group.

Ignorance of semidirect products and group extensions is quite understandable in a physicist, but also easily cured. These notes are an attempt to do so based on my experience trying to cure myself. They are self-contained, and the only prerequisite is a little group theory. Though I allude to topology in a couple of places, no knowledge of it is necessary.

These notes cover the following:

• Normal subgroups and quotient groups, the isomorphism thms, various other group-related concepts, and inner/outer automorphisms.
• Exact sequences, short-exact-sequences (SES’s), splitting of SES’s, central extensions, and the isomorphism classes of SES’s (as well as how they interact with splitting).
• Introduction to the hierarchy of direct products, semidirect products, and group extensions. We take 3 views of each: (1) an external view in which we build a new group from two distinct groups, (2) an internal view in which we consider the relationship between an existing group and its subgroups, and (3) an SES-view in which we frame things in terms of short exact sequences. Because the external-view is conceptually the most challenging, we give a preview of its regimen.
• The direct product in all 3 views.
• Semidirect products in all 3 views.
• General group extensions in all 3 views.
• A brief comparison of direct products, semidirect products, and group extensions in all 3 views.
• An addendum in which we explicitly construct the multiplication on a group extension in gory detail.

A few caveats. There are lots of detailed proofs in the notes. There also probably are typos and possibly errors. I hope to correct and clarify as needed, so please report these to me if you encounter any. The notes were written in passes and phases. There is a lot of intentional replication of effort in the proofs (where it serves the purpose of clarity), but also probably some unintentional replication (for example, if I consolidated some results in a proposition in one place, but fail to use those results in another). Nonetheless, I believe they should be quite informative and are on balance both correct and quite comprehensible. In particular, I tried to avoid “simplifying” things in subsequent passes as my understanding improved (though I certainly did correct things as needed). The danger in doing so is that I would lose sight of the things which confused me when first encountering the concepts — and one purpose of these notes is to address such sources of confusion. After enough passes and simplifications, the entire piece would end up a one page terse Bourbaki-like statement, along with a vague hint at the possibility of an idea of a proof-outline — which anyone worth their salt is expected to easily be able to expand into a 40-page proof. In math, the best way to learn a subject is to already know it. In physics, we take a gentler approach.

There are a couple of key omission — neither mission-central — which I hope to address in future posts. These concern (1) the relationship between semidirect products/group extensions and fiber bundles/principal bundles and (2) a detailed discussion of group cohomology and the classification of group extensions.

PDF Notes: Semidirect Products, Group Extensions, Split Exact Sequences, and all that