User:Vipul

I am Vipul Naik. I completed my B.Sc. (Hons) Math at Chennai Mathematical Institute and in Fall 2007, I began the Doctoral Program in Mathematics at the University of Chicago. I completed the program in December 2013. This wiki (on Group Properties) is my brainchild, and so far, I have been the principal contributor.

You can learn more about me at my personal website, vipulnaik.com.

Math formula tests:

$9^{\sqrt{7 + 2}} = 729$
$2^{7} - 1 = 127$
$2^{8 - 1} = 128$
$5^{1 + 2} = 125$
$\sqrt{7 + 2}!! + 4 = 724$

Notes for stuff I plan to expand

I'm putting notes here for stuff that I sort-of worked out but didn't want to start editing on the wiki.

Potentially characteristic subgroups characterization problem

Review tables in potentially characteristic subgroups characterization problem.

Characteristic not implies definable

New page characteristic not implies definable (or even more generally characteristic not implies second-order definable or even more general) that uses examples inside GAPlus(1,R) based on the fact that we can have really wild subfields between the rationals and reals, generated by any collection of transcendentals that are individually not definable numbers, but that the same argument that shows that the reals have no field automorphisms effectively makes all these intermediate subgroups characteristic.

Characteristic equals fully invariant in odd-order abelian group

Characteristic equals fully invariant in odd-order abelian group; the proof is correct, and makes the right logical moves, but it somewhat obscures what's going on. The key insight is that the compositions of direct factor inclusions and direct factor projections additively generate the endomorphisms, and all elements of this additive generating set have the property that adding the identity element to them gives an automorphism. The actual calculations to support this are done in the proof, but the bookkeeping of the nature of the subgroup itself, while interesting, somewhat detracts from the core point. Maybe an alternate proof that just relies on the (currently non-existent page) automorphisms additively generate ring of endomorphisms for odd-order abelian group makes the point, and the latter can hide the calculations.

3-subnormal and unique fastest ascending series

3-subnormal subgroup need not have a unique fastest ascending subnormal series:

Convert the second example family to tabular format
Work out numerical sizes of the smallest example based on the construction as described (for first construction, order 4 in order 2048?)
Compare with the theoretically smallest counterexample sizes possible (order 2 in order 32?)

Finite index implies powering-invariant

Finite index implies powering-invariant uses a proof that relies on going to the normal core, but there's a slicker direct argument that involves repeatedly taking $p^{t h}$ roots, noticing that you'll get overlap at some point, then working backward to get that the $p^{t h}$ root is actually in the subgroup. This is a more hands-on argument, and has both advantages and disadvantages over the presented argument.

Unfortunately I am having trouble completing the proof. Here's what I have so far. The challenge is that without the normality assumption, even when I get an overlap by the pigeonhole principle, taking powers on both sides requires left/right coset bookkeeping. Even though we can arrange for the overlaps to match up on both left and right cosets, it's still challenging to take powers. I'm sure this can be repaired, but it may be that the repair of this argument is messier than the abstract proof and therefore not as illuminative.

In fact, it would be interesting to get an explanation of why the proof strategy below cannot work without accumulating the full complexity of going to the normal core, and generically, going all the way to $n!$ for a subgroup of index $n$ . That would explain why we have to go down to the normal core.

Proof draft so far below:

Element-chasing version of proof

Given: A group $G$ , a subgroup $H$ of finite index in $G$ , a prime $p$ such that $G$ is powered over $p$ , an element $h \in H$ .

To prove: There exists an element $u \in H$ such that $u^{p} = h$ .

Proof: The tricky part here is that since $H$ is not normal in $G$ , we do not globally have a two-sided concept of cosets, and we cannot simply go modulo $H$ . However, since we are looking at roots of a single element, we do have commuting locally in the area of interest.

Step no.	Assertion/construction	Given data used	Previous steps used	Explanation
1	$h$ has a $(p^{n})^{t h}$ root in $G$ for any positive integer $n$ .	$G$ is powered over $p$ .		We essentially keep taking $p^{t h}$ roots repeatedly.
2	There exist distinct positive integers $a$ and $b$ with $a < b$ such that the $(p^{a})^{t h}$ and $(p^{b})^{t h}$ roots of $h$ are in the same left coset of $H$ .	$H$ has finite index in $G$	Step (1)	This is a simple pigeonhole principle argument for the map taking $n \in N$ to the left coset of $H$ of the $(p^{n})^{t h}$ root of $h$ .
3	Let $v$ be the $(p^{b})^{t h}$ root of $h$ from Step (2). Then, the $(p^{a})^{t h}$ root of $h$ is $v^{p^{b - a}}$ , and this is in the same left coset of $H$ as $v$ .		Step (2)	Step-direct

Finite subgroups leading to subgroups of finite index

Existence of finite normal non-central subgroup implies existence of proper normal subgroup of finite index: Use group action by conjugation on the normal subgroup.

Migrate answers from Math Overflow

Division in linear groups

General linear group over algebraically closed field of characteristic zero is divisible:

Add explicit formula for $n = 2$
Gesture at explicit formula for higher $n$
Add abstract argument covering higher $n$ ; basically, without finding the exact root, argue that taking the Jordan block for the root still works up to conjugation. The key is the claim that the block does not split into multiple blocks. The gotcha is that this argument must hinge on characteristic zero.
Discuss ambiguity when you have different blocks of the same eigenvalue
Reread Special linear group over algebraically closed field of characteristic zero is divisible precisely by those primes that do not divide its degree and update any ambient assumptions revised or refined

APS theory

Private Claude conversation

Wrapping up the finitely displaced, contrasaturated, etc. stuff

Contrasaturated of bounded defect, and rename existing result to point to that
Shortfall lemma that allows for converse given padding-saturation
Memberwise complemented normal sub-IAPS that combined with contrasaturated of bounded defect allows us to infer finite displacement

Asymptotics

Asymptotics, log log |G_n|, log log (#conjugacy classes(G_n)), relative to log n (Fekete), connect to Math Overflow problem on conjugacy classes and double logs. Lots of useful material in chat with Claude.

Automorphism APSes

Automorphism of polynomial rings, free groups, etc. Some useful material in chat with Claude.

Tight fit concept for APSes

For instance, generation by Levi subgroups. Search for tight fit in chat with Claude.

Various email write-ins

A few requests over email have been pending for a while. I should process them.

Structural and software fixes

MediaWiki upgrade

A MediaWiki upgrade is long overdue.

MathJax switch

I might want to try switching to MathJax once my MediaWiki version is at least 1.42, per https://www.mediawiki.org/wiki/Extension:Math#Native_MathML. This should make blackboard rendering work properly; this is a major issue right now because special formatting for reals and complex numbers and such blackboard characters is not working right now.

Semantic MediaWiki salvaging

The whole Semantic MediaWiki situation is a mess, with much of the desired benefit not happening -- the lists are not completely printed. This seems like it'll require some focus to investigate and fix.

Simple scripts to validate mathematical text and search for errors

Private Claude conversation with initial exploration

LLM assistance could help me write the parsing portions of these, but the ideas derive from my own lived experience. A few ideas:

Check for undeclared variables in long algebraic derivations. A simple "parse out the variables, check for undefined". This can live outside MediaWiki; it can just take a MediaWiki page entry or text snippet and run against that.
Check for obvious formatting issues like mis-spelled math tags.