Creative_math_

The wait is over. With great pride, the GAMMA Math Community presents to you the third edition of IGMO! Register in bio now!!!

IGMO saw massive success in its first two editions, with participation from 100+ countries and over 4000 participants, making it one of the biggest and most prestigious math olympiads.

This year, the game is at a whole new level. We have a prize pool of $2000 and a lot of opportunities and prizes for everyone! IGMO will be held in 2 rounds, starting from an easy-ish math contest and leading up to IMO difficulty levels. The registration has already begun! It is free of cost, and a credible participation certificate is guaranteed. Go to the the link in my bio and sign up now!

Follow our official account .info for all updates.

This deserves a long post, because I'd say it's probably the most important theorem in group theory (at least basic group theory). The first isomorphism theorem is fundamental not just to group theory, but also to other algebraic structures (there are very similar analogues for rings, modules, lie algebras).

It also gives us very cool results, and has a connection to the rank nullity theorem. This shows how fundamental this theorem is (heck, I'd have expected it to be called the fundamental theorem of group theory).

It also gives us the 2nd isomorphism theorem, which ends up being the basis of a way to classify finite groups using sylows theorems and semi direct products. Essentially, the 2nd isomorphism theorem gives us information about group products, and maybe if we're lucky sometimes, most finite groups will be a group product of its subgroups (spoiler, they are)

Finally, I end with some category theory ways to look at this. Its not very important, i guess it just gives an interesting framework to view it from. But it is cool that normal subgroups are EXACTLY kernels of homomorphisms! Initially it mightve felt like the definition of a normal subgroup was quite random, but its pretty natural.

Hope you enjoyed the post! Make sure to like and save. Also, I turn 20 today.





-----------------------------------------------------------
Follow
Follow
----------------------------------------------------------

In mathematics we often define the idea of a quotient object. Eg: in topology we have quotient spaces, in linear algebra we have vector space quotients, in group theory we have group quotients, etc etc etc..

Roughly speaking, taking a quotient means to treat some objects as though they are the same, and only deal with their representative (as long as this is well defined, ie, you really CAN only deal with the representative). A very good example is integers mod 3. If we take the group Z, the integers mod 3 are the set {0,1,2} where 0 is a representative of numbers like {..., - 3, 0, 3, 6,...} and 1 is a representative of {..., - 4, 1, 4,7,..} and so on. In this case, we really CAN just pick any representative and get the same answer. What i mean is, if i want to calculate - 3 + 4 mod 3, since - 3 has representative 0, and 4 has representative 1, i can simply do 0+1 and get the same answer (which is 1, which is right).

The normality condition is a condition for subgroups that lets us take quotients. We see where this condition comes from, consider some useful examples of quotients (center, kernel). We then show that normality isn't a transitive condition, and comment on sizes of quotient groups.






-----------------------------------------------------------
Follow
Follow
----------------------------------------------------------

This is part 5 of the group theory series, and in this we go over a very classic group, the group of symmetries of an n-gon.

D_n (or in some textbooks, D_2n for an n-gon) is an important example in group theory as it is always non-abelian and comes up as counterexamples to quite a few statements.

Also, D_n is important for classifying finite groups, which is what we intend to do anyways.

We cover its definition, how it's naturally a subgroup of S_n, calculation of its size, what its element all look like and finally we find its center. Hope you like the post, do like and save/






-----------------------------------------------------------
Follow
Follow
----------------------------------------------------------

I was mentioning that we will prove number theoretical results using group theory, and here is one. A large portion of early group theory (before people even agreed on what a group was) was done from a number theoretical context.

This theorem simply falls apart from a group theory result. I mean, really, it's just a 2 line proof. I just decided to explain some terminology and write out what I was saying clearly, but it's an extremely short proof.

Fermat's little theorem (a big one in Olympiad math) simply falls out from this as a corollary as well. I like that kind of stuff. It's a cute, easy to see corollary.

Euler's theorem is the basis of the RSA encryption algorithm (I've made a post on this before) so it's quite important, and we see the power of group theory already with such a simple proof.

There will be a YouTube video out soon where I prove more number theoretical results using group theory. Stay tuned for that, there is a lot of fun stuff to go over.






-----------------------------------------------------------
Follow
Follow
----------------------------------------------------------

There's a lot to go over in this post honestly, but the main point I want you to take away is that an isomorphism means that two groups are the same from a group theoretical perspective. That means all of its group properties are the same. If you gave someone(who didn't know what the symbols in your groups meant) your two cayley tables, they wouldn't be able to tell the difference.

The post goes over a lot, we cover homomorphisms, isomorphisms, and what the cayley table of a group is. I try to really explain what it means for 2 groups to be the same.

For group theory purposes, if two groups are the same we only study one of them (cuz it tells us everything we need to know about the other). We now shift our attention towards classifying groups. How wacky can groups get? Clearly, groups like S_3 and Z_6 are different, but how *different* can groups of order 24 really get?

Finally, we end with classifying all groups of order p, where p is prime. Surprisingly, each such group must be isomorphic to the group {0, 1, 2, 3,..., p-1}, which is the group Z_p under addition mod p.

I hope you liked the post! We are now ready to give quite a few applications of group theory to number theory.





-----------------------------------------------------------
Follow
Follow
----------------------------------------------------------

Lagrange's theorem is one of the most important theorems in finite group theory. Simply speaking what it says is: the order of a subgroup divides the order of the group.

This may not look like much, until you pause and think... Wait the order of ANY subgroup MUST divide the order of the group? So what if I have a group of order 6? It's subgroups can only have size 2 and 3? (other than 1 and 6 ofc, but those are the obvious subgroups). The answer is yes, it can only be of size 2 or 3. If you have a prime order group, it has NO subgroups other than the trivial one and itself.

We haven't covered order of an element yet but when we do, Lagrange will come in yet again and we will use this to prove a few number theory facts.

What I want you to take away from this post is an important fact: cosets partition our set, i proved this in 2 ways and the 2nd one is part of something more general (they form an equivalence class, look this up if you don't know it yet, this is central to mathematics). This will generalize nicely when we cover the orbit stabilizer theorem.

Besides, I just tried to prove the theorem step by step in an easy manner, with examples provided to help sink the lemmas in. Hope you enjoy the post! We are technically, now, able to classify all groups of prime order (once we define what all those words mean) so yay!






-----------------------------------------------------------
Follow
Follow
----------------------------------------------------------

In this post we introduce the basics of group theory. The definition of a group, along with several examples of groups that we will find useful along the way. We also introduce the notion of a subgroup and give a few examples of it.

Yes, it is real. I'm making a series on group theory. This is the first post. It'll be an introduction to group theory at the early undergraduate level (though even highschoolers would be able to understand most of it) with an emphasis on finite group theory and classifying groups of finite order.

We will also focus on number theoretical applications of group theory (there are quite a few) and some combinatorics you can do with group theory. I intend to use the concept of group actions quite a bit.

Groups are a very natural construct, we will give several examples of them throughout the series but mostly we will prove results in generality, using the axioms/definitions given. I hope you enjoyed this post and found it easy to understand.






-----------------------------------------------------------
Follow
Follow
----------------------------------------------------------

Every non-constant polynomial of degree n with complex coefficients has n complex roots counting multiplicity. This statement is the fundamental theorem of algebra, and is often swept under the rug in linear algebra courses.

In this post though, you shall see it being proved using the tools of algebraic topology. The proof is borrowed from Hatcher's textbook on algebraic topology.

There is something interesting about this theorem though. It is not fundamental to algebra. If you were to ask me what this is fundamental to, I'd say it should be called the fundamental theorem of C. Because what it's really saying is that C is an algebraically closed field.

Thinking a bit more about this, we know the reals aren't algebraically closed. The polynomial x²+1 has no real roots. However, if we were to consider 2nd degree polynomials, and "extend" R so that every 2nd degree polynomial with coefficients in this extension also has roots in this extension, we know by the quadratic formula that simply adjoining i will do it. That is to say, R[i] = {a + bi : a, b € R} will "complete" R at least for 2nd degree polynomials. What's interesting, is that this completion is all you need for any arbitrary degree! Essentially, that's what FTA is saying. I think it's a nice way to look at it, one would perhaps expect that allowing completion upto higher degree polynomials would mean you have to extend your field more, but for the case of R, simply completing up to 2nd degree means you complete it for all degrees. Hope you liked this post!






-----------------------------------------------------------
Follow
Follow
----------------------------------------------------------

Here's a mathematician I haven't posted about before! Paul Erdős is regarded as the greatest problem solver of the 20th century. He's got the highest number of papers published out of ANY mathematician to ever exist, with the number of papers he's published exceeding 1500. Most of his work was in number theory and combinatorics, and this is one of his works.

This was actually his FIRST paper and he published this remarkable proof at the age of 19 (!!!). Here I am at 19 just admiring how beautiful this proof is, one could clearly guess that the person who came up with this was phenomenally talented.

Many of his conjectures still remain unsolved and are still the topic of active research.

I hope you guys enjoy this post, make sure to like and save it!

Here's a post on Legendre's formula! (the legendary formula)

Fun fact : I was actually supposed to post about something else and that needed this formula, but I asked on story and most of you responded that you hadn't seen it. I asked whether he's covered a proof of it before (cuz his posts are epic and it would've worked as a good reference) but he'd only posted the formula, not the proof.

So here is its proof (quite a short one honestly) and more relations this formula has with base p stuff (I prove a nice formula about its relation to sum of digits in base p). The proof I give is in my opinion better than the one on Wikipedia.

Another fun fact : yea, Legendre looks weird. I know. However, there is no photo of him on the internet other than this! More interestingly, a WRONG photo was attributed to him until 2005, when the mistake was discovered and it turned out that it was a guy named Legendre, but not the mathematician. I doubt this painting is accurate, he looks like a ghost here.

The formula is also known as de polignac's formula, however there's no picture of de polignac on the internet either (why didn't he just post a selfie on his Instagram???)

I hope you enjoyed, more number theory coming soon.

-----------------------------Follow
Follow
Follow
-----------------------------

Let's talk about some set theory today.

We know how to define cardinalities of finite sets, that's easy, just count the number of elements. What about the cardinalities of infinite sets? I mean, they're still sets, and I guess they do have a "size" so how would we compare their sizes?

In comes cantor, we simply take a more generalizable approach. We can also say the cardinalities of two finite sets are the same if there's a bijective function between them, so we can simply extend this to be the case for infinite sets too.

However, do the things we expect to hold for finite sets hold for infinite sets when it comes to cardinalities too? Not always! But yes, sometimes, and this theorem is that one sometimes. Essentially, if |A| >= |B| and |B|