# A Brief Tour Through Basic Group Theory – Part 4

by Anna Mustata

Part IV – Relating Groups to Other Groups

Attempting to make things more concrete leads us to developing the concept of group actions. We can now use this to zoom back out. We’ve been applying groups to geometric objects to find out how they can tell us more about the things we we’re already interested in, but can we use this to make them tell us more about themselves?

Of course, I wouldn’t be asking this rhetorical question unless the answer was yes.

Before we try to apply groups to themselves or to other groups, let’s establish a general way to talk about relations between groups. To see how we should look at these relations, we return to the key principle of keeping track of information. When we work in a very concrete setting – say, adding integers or looking at symmetries of a cube – the set associated to our group is very important. At a more abstract level, however, this information becomes irrelevant; the structure of the group is not determined by what the elements of the set are, but by how they combine with each other. If we were to write out the multiplication table (that is, a list of which element is formed by every possible combination of two other elements) for symmetries of the cube, then replace each function with a Greek letter everywhere it occurs, we would call the second group isomorphic to the first. That is to say, they are essentially the “same” group, but may represent a different concrete situation.

A more interesting relation is the homomorphism. Like an isomorphism, a homomorphism equates elements of the underlying set of one group to elements in the underlying set of another without changing the multiplication table – that is, if a goes to a’, b to b’ and c to c’, then a*b = c means a’*b’ = c’. The difference is that multiple elements from one group can be equated to the same element in the second, and not every element in the second group must be reached.

We could say that equating different elements in the first group to a single element in the second group is “gluing” them together, in the same way we glued functions together when performing a group action. The functions from the first group which are glued to the identity element from the second now perform the same role that a stabilizer performed for a group action. In this case, we call it a kernel. The elements “glued” to any other element in the second group are cosets of the kernel.

We can imagine this gluing as “folding” the landscape of a group so that all elements in the kernel are brought to the “starting point” of the landscape. It makes sense to wonder how other symmetries are affected by this folding. In particular, we’ll look at the conjugacy symmetry where all similar paths radiate out from the starting point. If we consider the kernel as one of these paths, bringing it into the starting point requires by symmetry that we must bring all similar paths into the starting point as well. However, the kernel is defined as containing all elements that get brought into the center, so we must conclude that there are no other similar paths; in other words, conjugating the kernel by any other element in the original group returns the kernel. Subgroups which have this property are known as normal subgroups.

Returning to the concept of groups acting on groups, we will see that examples of this have already snuck up on us and are fundamental to our descriptions of symmetries in groups. In our first symmetry, we “stepped off the path” of a subgroup by applying a function belonging to the larger group, but not necessarily the subgroup. This was, in fact, the larger group acting on the subgroup (or rather, on all cosets of the subgroup). The orbit was the set of all subgroups, and the stabilizer was the set of all elements in the subgroup itself. The same can be said for conjugating a subgroup. When that subgroup is normal, its orbit is itself and its stabilizer the entire group containing it.

At this point, the concept of a homomorphism and the concept of a group action may look rather similar. That’s because a group action is a homomorphism. When we “apply” a function to a set, we are choosing a way for the elements of that set to be reordered amongst each other. The set of all possible such reorderings is, itself, a group known as the symmetric group of order n, or Sn, where n is the size of the set that is being reordered. Sn is an incredibly versatile group, since it can describe the reorderings of any set of size n. Our group action on a set of size n is then in fact a homomorphism from the group that acts on the set to the group Sn.

We’ve discussed how a group can act on a subgroup of itself. However, observe that a group is a subgroup of itself. We define the regular action of a group on itself where applying an element of the group to another element of the group is the same as combining them by the group operation. This may seem like a pointless thing to do, but looking at the previous paragraph, we see that it gives us a homomorphism from any group to a symmetric group Sn where n is the size of the first group. If we examine this homomorphism, we see that its kernel contains only the identity element. In other words, no information is lost by gluing things together during the homomorphism; the group stays intact, so to say, and is merely placed directly over some path in Sn. Therefore, we can say that any group of size n is isomorphic to a subgroup of Sn. This is called Caley’s theorem.

At this point, we have gone from thinking a group can be “basically anything” to seeing all groups of finite size as subgroups of very specific types of groups, with properties that can be understood by a person with some knowledge of permutations.

There is one more thing we want to discuss in this section. So far, we’ve used homomorphisms to compare two groups we already knew about. However, we can also create new groups like this. We’ve already discussed “folding” over a kernel. Any normal subgroup can be used as a kernel; the process of folding this subgroup into the starting point is known as quotienting the larger group by the subgroup. Each coset of the subgroup becomes a single element in the new group. Anyone familiar with modular arithmetic can see that if we were to quotient the group of all integers by the subgroup of even integers described at the very beginning of this article, we would get a group with addition in mod 2.

If Caley’s theorem allows us to narrow down what groups can be by embedding them in larger groups, quotienting allows us to simplify groups by breaking them down into pairs of smaller groups: a normal subgroup and its quotient group. A group that can’t be broken down any further is called a simple group (remember, these are the ones that have been classified in the Atlas of Finite Groups). Simple groups play a similar role in group theory as primes play when working with integers.

Part V – Conclusion

At this point, you’ve seen most of the essential ideas that allow you to work with groups. So, what now?

On the concrete side, we can use these concepts to study particular groups that are of natural interest. Some such groups have been mentioned already. Symmetry groups of geometric objects tell us things about how molecules behave in chemistry and quantum mechanics. If we restrict this symmetry to rotations in two dimensions, we get complex number multiplication which allows us to find roots to equations of the form xn = 1. Groups of polynomials in multiple variables allow us to study the symmetries of these polynomials, which in turn allows us to solve polynomials in one variable of degree greater than two by looking at the symmetry of its roots. Groups of invertible matrices allows us to look at geometry – more specifically, transformations in geometry – using a simple set of algebraic rules, at a higher level than we can reach just with trigonometry and cartesian co-ordinates. The example given with colouring the faces of a cube shows that group theory can be used to solve combinatorical problems. Finally, modular arithmetic (the groups obtained by quotienting the integers by a subgroup) is very useful in cryptography.

On the more abstract side, we now just about have the tools to start categorizing the types of groups that can exist. Based on what’s been discussed in this article, it should be possible to categorize all groups of size up to eight. Categorizing groups of larger size requires some more theorems, but still follows essentially the same principles.

## 99 thoughts on “A Brief Tour Through Basic Group Theory – Part 4”

1. Hello. remarkable job. I did not expect this. This is a impressive story. Thanks!

2. I was recommended this website by my cousin. I am not sure whether this post is written by him as nobody else know such detailed about my problem. You are wonderful! Thanks!

3. Hi, just required you to know I he added your site to my Google bookmarks due to your layout. But seriously, I believe your internet site has 1 in the freshest theme I??ve came across. It extremely helps make reading your blog significantly easier.

4. Amazing blog! Is your theme custom made or did you download it from somewhere? A theme like yours with a few simple adjustements would really make my blog shine. Please let me know where you got your design. Thank you

5. Wilburnouse says:

buy cialis very cheap prices fast delivery: buy cialis online viagra much does cialis cost without insurance
cialis for daily use

6. Perfect work you have done, this site is really cool with wonderful info .

7. Thank you for the auspicious writeup. It in fact was a amusement account it. Look advanced to far added agreeable from you! By the way, how could we communicate?

8. Of course, what a fantastic blog and illuminating posts, I surely will bookmark your site.Best Regards!

9. amoxil suspension refrigeration para que sirve amoxil 500 mg how to store amoxil suspension before reconstitution

10. ulan herkes tokatçi olmus ama bu adam gönderdi helal valla??

11. You made some decent points there. I seemed on the web for the difficulty and found most people will associate with along with your website.

12. ivermectin rabbit dosage ivermectin virus ivermectin dosage for dogs heartworm prevention guinea pig dighting me when i put on ivermectin

13. milbemycin and ivermectin ivermectin 3 ivermectin for demodectic mange in dogs gaba receptor in border collies affected by ivermectin inhibit what ion

14. ivermectin egg withdrawal ivermectin 250ml can i put ivermectin on my cat why do my hands turn purple treating with ivermectin

15. molnupiravir about pill that covid19 hospitalization molnupiravir uk molnupiravir about pill that cuts hospitalization molnupiravir pharmacy

16. Of course, what a magnificent website and educative posts, I surely will bookmark your blog.Best Regards!

17. I loved as much as you’ll receive carried out right
here. The sketch is attractive, your authored subject matter stylish.
nonetheless, you command get bought an nervousness over that you wish be delivering the following.
unwell unquestionably come more formerly
again as exactly the same nearly very often inside case you shield this increase.

18. What’s up, its nice article concerning media print, we all be
aware of media is a fantastic source of information.

19. Hi there just wanted to give you a quick heads up. The words in your content seem to be running off
the screen in Firefox. I’m not sure if this is a format issue or something
to do with web browser compatibility but I figured I’d
post to let you know. The style and design look great though!
Hope you get the issue resolved soon. Thanks

20. Wow that was odd. I just wrote an incredibly long comment but
after I clicked submit my comment didn’t show up.
Grrrr… well I’m not writing all that over again. Regardless, just wanted to
say fantastic blog!

21. This article gives clear idea designed for the new people of blogging, that in fact how to do blogging and
site-building.

22. I have been exploring for a little bit for any high quality articles or blog posts in this kind of space .
Exploring in Yahoo I eventually stumbled upon this
web site. Studying this information So i am glad to show that I have a very good uncanny feeling I discovered just
what I needed. I so much definitely will make certain to do not omit this web site and provides it a glance on a relentless
basis.

23. Hello, i believe that i noticed you visited my website so i came to go back the want?.I am trying to in finding issues to improve
my site!I assume its good enough to use some of your ideas!!

24. My brother suggested I might like this website. He used to
be entirely right. This submit actually made my day.
You can not consider just how much time I had spent for this information! Thank you!

25. Heya this is kinda of off topic but I was wondering if blogs use WYSIWYG editors or if
you have to manually code with HTML. I’m starting a blog soon but have no coding
know-how so I wanted to get advice from someone with experience.
Any help would be enormously appreciated!

26. I’d like to find out more? I’d like to find out more details.

27. I was recommended this web site through my
cousin. I’m now not certain whether this put up is written through him as nobody else
know such detailed about my trouble. You’re wonderful! Thank you!

28. veifs says:

** Talletus kortilla suoraan **. Useimmilla kasinoilla (mutta ei kaikilla) on sopimuksia luotto- ja maksukorttiyritysten kanssa, kuten Visa, Mastercard, American Express, Diners ja Maestro. Jos he tekevät niin, kasino pystyy käsittelemään maksusi suoraan ilman ylimääräistä vaihetta käydä e-lompakon läpi välittäjänä. Suomalaiset ovat kovaa pelikansaa ja siksi voitkin löytää laajan valikoiman erilaisia kasinoita, jotka palvelevat suomeksi. On olemassa täysin suomalaisia kasinoita, mikä tarkoittaa sitä, että taustalla on joukko suomalaisia eksperttejä työskentelemässä, kasinon nimi on suomeksi, sekä koko nettikasino palveluineen on tehty suomalaisia varten. Tällaisia kasinoita voit löytää jatkuvasti yhä enemmän, sillä suomalaiset ovat yksi tärkeimmistä asiakkaista nettikasinoiden maailmassa. https://waseemjutt.com/demos/pnp/community/profile/starhann5772892/ 1. Casumo 2. Guts 3. Rizk 4. Wildz 5. Casino Winner 6. CasinoEuro 7. LeoVegas 8. Katso kaikki Siirto kasinot. Valitettavasti Siirto on tällä hetkellä saatavilla ainoastaan yläpuolella mainittujen pankkien asiakkaille. Uskomme kuitenkin, että tulevaisuudessa sitä pääsee käyttämään muutkin. Emme kuitenkaan voi sanoa tätä varmaksi, mutta toivotaan parasta! Jos et vielä tiedä millä sivustolla voit käyttää Siirto -maksua, tsekkaa tämän hetken parhaat Siirto kasinot näppärästi alapuolelta löytyvästä kattavasta listasta. Siirtomaksu toimi kätevästi. Kun haluat tehdä kasinotalletuksen Siirtoa käyttämällä, tulee kasinolla tietysti olla Siirto maksuvaihtoehdoissa, eli ilman sitä se ei ole mahdollista. Siirto on yleensä kuluttajille maksuton, mutta koska Siirto-sovelluksia tarjoavat pankit ja maksupalvelut määrittelevät itse hintansa, kannattaa hinnat aina tarkastaa. Kasinoiden omat talletuskulut kannattaa myös aina tarkastaa.

29. Hello! I understand this is somewhat off-topic however I needed to ask.
Does managing a well-established blog such as yours
take a massive amount work? I’m brand new to writing a blog but I
do write in my diary on a daily basis. I’d like to start a blog so I can easily share my experience and views online.
Please let me know if you have any kind of suggestions or tips for brand new aspiring bloggers.

Thankyou!

30. Wow, fantastic weblog structure! How long have you been running a blog for?
you make running a blog glance easy. The whole glance of your website is fantastic, as smartly as the content!

31. I do consider all the ideas you’ve introduced for your post.

They are very convincing and will certainly work. Still,
the posts are too brief for newbies. Could you please lengthen them
a bit from next time? Thanks for the post.

32. You really make it seem so easy with your presentation but I
find this topic to be really something that I think I
would never understand. It seems too complex and very broad
for me. I am looking forward for your next post, I’ll try to get the
hang of it!

33. Unquestionably believe that which you said.

Your favorite justification seemed to be on the web the simplest thing to be aware of.
I say to you, I certainly get irked while people think about worries that they plainly don’t
know about. You managed to hit the nail upon the top and defined out the
whole thing without having side effect , people could take a signal.
Will probably be back to get more. Thanks

34. Hi there it’s me, I am also visiting this website daily, this website is really fastidious and the viewers are really sharing nice thoughts.

35. We stumbled over here from a different page and thought I might as well
check things out. I like what I see so i am
just following you. Look forward to going over your web page again.

36. Appreciating the persistence you put into your blog and
detailed information you provide. It’s great to come across a blog every once in a
while that isn’t the same out of date rehashed information. Great read!

37. great put up, very informative. I ponder why the opposite experts of this sector do not notice this.
You should proceed your writing. I am confident, you’ve a huge readers’

38. Unlah says:

39. Hi, just wanted to say, I liked this blog post. It was funny.

Keep on posting!

40. Outstanding quest there. What occurred after?
Thanks!

41. Hi there, its pleasant post on the topic of media
print, we all be familiar with media is a great source of facts.

42. Simply want to say your article is as astonishing. The clearness in your post is simply spectacular and i
could assume you are an expert on this subject. Well with your permission allow me to
grab your feed to keep updated with forthcoming post. Thanks a million and please
continue the rewarding work.

43. nem says: