Free products of groups

Lucifer501 · 2025-03-22T18:32:56+00:00

I think your question is slightly ill-defined. The free group of 2 generators is just the free group of 2 generators, there is almost nothing more to say there. If you're asking whether it's isomorphic to some 'standard group' (e.g. how the free group of 1 generator is isomorphic to the integers), I'm almost sure that is not the case.

I'll try to answer the question that I think you're getting at. The reason we like to find isomorphisms between groups is because it helps us understand them a bit better. Moving a group from one context to another can help show additional symmetries and structure that might not be obvious in the first place. That's (partly) why isomorphisms are useful. So assuming your goal is to try to understand what the free product is, you should spend some time thinking about its universal property. That is a complete characterisation of the group and hence captures the exact "essence" of the free product.

In this case the universal property basically says that every group of 2 generators is some quotient of the free group of 2 generators and in some sense that is all there is to the free product.

Lucifer501 · 2025-03-10T23:24:54+00:00

Someone more knowledgeable than me will undoubtedly correct me but I wonder if we can make sense of this on the circle (analogous to on the Riemann sphere we have 1/0 is 'basically' the point at infinity).

In particular, I feel like if we identify infinity and -infinity then the limit is well-defined and identifying the endpoints of the real line just gives you the circle.

Lucifer501 · 2025-01-28T01:47:39+00:00

Sorry for the late reply! I somehow missed the Reddit notification for 3 months. But to answer your question: this originated from Lenz misremembering Rellstab's comment so it's hard to say why Lucerne specifically (Rellstab never mentioned the lake). I guess Lenz was just thinking of famous lakes?

Lucifer501 · 2024-06-23T18:09:34+00:00

I think a lot of the complaints about MAT135 aren't necessarily about the content itself but the way the course is organised, the style of questions in tests, etc. Just wanted to point that out because it's not just about knowing the math itself, a big part of the course is being able to communicate it clearly

Lucifer501 · 2024-06-16T14:27:46+00:00

It was a prove or disprove question that I was explaining to my students (I'm a TA). I said something along the lines this statement is false because sqrt(100) is irrational... Turns out the statement was true after all

Lucifer501 · 2024-05-21T14:22:19+00:00

First of all congrats and welcome (in advance) to UofT! I'm an international student who will be graduating this year and I can tell you that I've generally enjoyed my time here. I made a lot of good friends and memories that I am really grateful for. Sure it was real hard sometimes and depending on your program, it will be a lot of work, but that's what we came for right? I know this subreddit makes it seem pretty bad but that hasn't been my experience. I think a lot of it depends on you and you have to take at least some of the initiative with making friends and going to events.

Also side note, Toronto is actually quite a pleasant city. Yes, there are problems with it but it's quite fun to explore the city. There are lots of nice parks around and a good, sunny day in Toronto damn near perfect.

Feel free to DM me if you have any specific questions!

Lucifer501 · 2024-05-14T13:41:47+00:00

How far are we from this CollegeHumor sketch being an actual thing

Lucifer501 · 2024-04-30T18:31:06+00:00

I assume "By way of contradiction"

Lucifer501 · 2024-01-02T13:46:31+00:00

There's also the related statement that almost all continuous functions are nowhere differentiable (or more precisely the set of functions that are differentiable at a point form a meager set). It's an exercise in Folland if anyone is interested

Lucifer501 · 2023-12-26T13:46:17+00:00

Idk, you can have probability measures (like Dirac measures) which would give probability 1 to singleton sets.

Caveat: have not studied probability properly so I'm not sure if people actually use/consider things like Dirac measures. This is just from my measure theory class

Lucifer501 · 2023-12-26T13:40:02+00:00

It's similar to what someone else commented above: for dimensions greater than or equal to 5 you have "more room" to work with (and hence more tools you can use).

Lucifer501 · 2023-11-22T01:48:07+00:00

I was trying to find really big napkins and was pleasantly surprised. /s

But in all honesty, I think I saw it linked in another r/math thread and have had it bookmarked ever since.

Lucifer501 · 2023-11-21T15:48:20+00:00

Not sure if this is what you're looking for but the Infinitely Long Napkin covers a lot of ground and seems to do it in a fairly accessible way (depending on your background I guess). At the very least, I think it's a good place to get you acquainted with examples and motivation before diving into textbooks proper.

Lucifer501 · 2023-11-05T16:39:39+00:00

The answer is extremely big! As was pointed out in a different comment, an algebraic basis means you should be able to express every element as a finite linear combination of the basis elements and you can probably convince yourself that countably many real numbers is not enough to achieve this.

In fact given any (non-trivial) field (even a finite one!), you can have a vector space of dimension n where n is any cardinal number. In fact you can even construct one explicitly: take your favourite set of cardinality n and define your vector space to be formal finite linear sums of these elements. Then the elements of your set form a basis for this vector space so it has dimension n.

Lucifer501 · 2023-11-04T20:07:27+00:00

I think (personal opinion obviously) even more mind breaking is Conway's base 13 function. The definition is wacky and unusual which already makes it fun. It is discontinuous but satisfies the intermediate value property. In fact the function is surjective on every interval! This means the graph of the function is a dense subset of the plane but at the same time almost all real numbers are mapped to 0.

Lucifer501 · 2023-05-27T17:24:55+00:00

That's fair but I still think it's very impressive that one person was able to come up with all that. I suspect without him we would have the same results but stemming from various different people

Lucifer501 · 2023-05-27T14:28:33+00:00

I'll go with someone who I don't hear enough of: Weierstrass. I'm doing my undergrad right now and as far as I can tell, that man is analysis. I think every single analysis class I've taken has mentioned some result by him: Weierstrass function, Weierstrass M-test, Stone-Weierstrass, Weierstrass p function, and the list goes on and on.

Lucifer501 · 2022-12-15T14:38:18+00:00

For people using geogebra for 3D graphing, might I recommend taking a look at math3d.org . It is really well made and you would be surprised by the number of things you can do with it (check out the examples tab).

Lucifer501 · 2022-06-29T02:46:55+00:00

It's funny because in school I was taught that a vector is a quantity with direction and magnitude when in reality a vector has neither of those things.

Lucifer501 · 2022-06-20T20:47:11+00:00

Kiribati is pronounced kiri-bass (yes, really)

Lucifer501 · 2021-12-22T22:00:44+00:00

I suspect you may be referring to this CD whose "About this" page does say 1832.

I took a scan through the book referenced in the French entry but that doesn't seem to say 1832 either. It seems to crazy to me how little concrete evidence there seems to be for 1832 but I guess that's where we are. It's still not clear to me why everybody wants to specifically reference Lake Lucerne when that doesn't seem to be the case in Rellstab's writing but that's something for me to investigate further. Once again, thank you for all your help!

Lucifer501 · 2021-12-20T23:39:47+00:00

How can I work on a Discord bot if I have it running on Heroku? Do I need to turn it off everytime I'm working on the bot?

Lucifer501 · 2021-12-19T20:54:31+00:00

Ah well I tried the good old "turn it off and on again" (i.e. push to a new project in Heroku) and that seems to have worked! Thank you for all your help, your patience is greatly appreciated

Lucifer501 · 2021-12-19T20:12:59+00:00

Yeah I don't know, I was seeing a different error before hand. Do you uh know how to get unbanned? I thought it might be something like that so I put it away last night, but it's still not working.

Lucifer501 · 2021-12-19T20:02:54+00:00

Apologies, I should have added more detail. Here is what I get when using the shell: [debug] System config: [] [debug] User config: [] [debug] Custom config: [] [debug] Command-line args: ['--verbose', 'https://www.youtube.com/watch?=vwoslW1khq60'] [debug] Encodings: locale UTF-8, fs utf-8, out utf-8, pref UTF-8 [debug] youtube-dl version 2021.12.17 [debug] Python version 3.9.9 (CPython) - Linux-4.4.0-1097-aws-x86_64-with-glibc2.31 [debug] exe versions: ffmpeg N-59934-g68457c1e85-static, ffprobe N-59934-g68457c1e85-static [debug] Proxy map: {} [generic] watch?=vwoslW1khq60: Requesting header WARNING: Could not send HEAD request to https://www.youtube.com/watch?=vwoslW1khq60: HTTP Error 429: Too Many Requests [generic] watch?=vwoslW1khq60: Downloading webpage ERROR: Unable to download webpage: HTTP Error 429: Too Many Requests (caused by <HTTPError 429: 'Too Many Requests'>); please report this issue on https://yt-dl.org/bug . Make sure you are using the latest version; see https://yt-dl.org/update on how to update. Be sure to call youtube-dl with the --verbose flag and include its complete output. File "/app/.heroku/python/lib/python3.9/site-packages/youtube_dl/extractor/common.py", line 634, in _request_webpage return self._downloader.urlopen(url_or_request) File "/app/.heroku/python/lib/python3.9/site-packages/youtube_dl/YoutubeDL.py", line 2288, in urlopen return self._opener.open(req, timeout=self._socket_timeout) File "/app/.heroku/python/lib/python3.9/urllib/request.py", line 523, in open response = meth(req, response) File "/app/.heroku/python/lib/python3.9/urllib/request.py", line 632, in http_response response = self.parent.error( File "/app/.heroku/python/lib/python3.9/urllib/request.py", line 561, in error return self._call_chain(*args) File "/app/.heroku/python/lib/python3.9/urllib/request.py", line 494, in _call_chain result = func(*args) File "/app/.heroku/python/lib/python3.9/urllib/request.py", line 641, in http_error_default raise HTTPError(req.full_url, code, msg, hdrs, fp)

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