doesn't mean !
11 Oct 2020
I spend a lot of time on math stackexchange (mse), and I periodically see “simple” questions that totally shatter a misconception that I didn’t know I even held.
My first time experiencing this was when I learned that normal subgroups need not be transitive. That is,
If
For cultural growth, we do have transitivity whenever
In fact, not only is
we also have a “weak transitivity”
I forget exactly when I first saw that normal subgroup-ness isn’t transitive, but I remember it was shocking. Even though in hindsight it is obvious how normalcy fails to be transitive, it was never something I explicitly saw. Moreover, the notation is so suggestive of a transitive relation, it felt more “obvious” than it probably should have. I had this realiziation long enough ago that I no longer have the mse post where I first saw it. However, I have had two similar realizations in recent memory, and I figured I would include them here as quick posts analogous to my “quick analysis trick” series.
The first fact is that simple groups
need not be hereditary. That is, if
This misconception is interesting to me because I think if you asked me “are simple groups hereditary?” I would have probably answered “I don’t think so?” as a gut reaction. Giving myself the benefit of the doubt, I’ll say I might even have come up with a counterexample after a minute or two.
Even though I think I would have said “no” when asked, when I first saw this written out it struck me as somewhat surprising. At the very least, it struck me as a mistake a younger me might have made. Even knowing it is false, though, I don’t think I would not have guessed how badly it fails! Again, I’m afraid I’ve lost the mse link to the question where I learned about this, but I remember the idea:
Every finite group embeds into some alternating group
We know by Cayley’s Theorem that every
Then every
Once we know that every finite group embeds into some alternating group, it is natural to ask the same of other families of finite simple groups. This question has been asked before, and there are references on that post which show that some natural families of simple groups don’t have this property (such as the Suzuki Groups), but I don’t know enough about the classification to properly think about this myself. Someday, though!
The second fact is one that I saw today, in this question.
There are abelian groups
This caught me super off guard. Unlike the last one, my gut reaction would
be that whenever
Here we have
One natural question is “when does
-
Enumerate the (finitely many) groups of order
(up to isomorphism) as . Then we can simply product them all together to get a (rather large) finite group . This is obviously inefficient, but it seems like not much is known about how small we can make . ↩