# Abelianness and equivalent definitions in general

In group theory, we say that a group $(G, \cdot)$ is abelian if (and only if) $a \cdot b = b \cdot a$ for all $a, b \in G$.

Abelian groups have certain properties that are generally (that is in the mathematical sense – meaning always) true, so if we know that a group we are dealing with is abelian, we can instantly deduce a variety of things about the group and how it behaves.

Once we know a group is abelian, we can make some useful deductions. Does it work the other way around? Are there truths that will allow us to conclude that a group is abelian?

For what statements are both of these true? What statements imply, and are implied by, the fact that a group is abelian? Essentially, we are asking:

From here onwards we’ll use juxtaposition (like $ab$) rather than an explicit symbol (like $a \cdot b$) to indicate the group operation acting on elements $a$ and $b$.

As an example, take $$(xy)^2 = x^2 y^2 \;\forall x, y \in G$$ Clearly, if G is abelian this is true. Why? Because $$(xy)^2 = (xy)(xy) = x(yx)y = x(xy)y = (xx)(yy) = x^2 y^2$$ using the fact that the group operation is associative and $G$ is abelian.

However, if we write out our first condition slightly differently

$$xyxy = xxyy$$

and then apply the ‘cancellation property’ of groups (cancelling the leftmost $x$ and the rightmost $y$), we end up with

$$yx = xy$$

Succinctly, we have proven

$$(xy)^2 = x^2 y^2 \iff xy = yx$$

for arbitrary elements $x,y$ of a group $G$. Since the elements are arbitrary, we have proven that $G$ is abelian if and only if our original condition is true.

In fact, we might as well define a group to be abelian if it satisfies this property. It may seem strange, but this doesn’t change anything about the mathematics at all. Which definition of the long (and in some sense infinite) list of equivalent definitions we happen to choose to be ‘the definition’ is an entirely human distinction. We can take whichever one we like, and the others are then just consequences of this definition (theorems in our theory).

How about another example? Define the direct product of groups $(G, \circ_{G})$ and $(H, \circ_{H})$, denoted $G \oplus H$, to be the cartesian product $G \times H = \{ (g,h) : g \in G, h \in H \}$ with the group operation $\circ$ defined such that $(g_1, h_1) \circ (g_2, h_2) = (g_1 \circ_{G} g_2, h_1 \circ_{H} h_2)$.

Clearly, the direct product of some number of cyclic groups is abelian, since the cyclic groups themselves are abelian, and the componentwise multiplication of the direct product reduces abelianness in the product to abelianness in each participating group.

It is a much deeper fact that the converse is true too: every (finitely generated) abelian group, say $G$, is isomorphic to the direct product of some number of cyclic groups: this is the so-called Fundamental Theorem:

$$G \cong \bigoplus_{i=1}^{n} \mathbb{Z}_{k_i}$$

where $k_1, k_2, \dots, k_n$ are prime powers.

Again, we might as well define an abelian group to be one that has this form, and then the fact that an abelian group’s elements all commute with each other is just another provable theorem.

I think this subtle and perhaps surprising notion of equivalence is rather interesting. It is conceptually easy to prove that two statements are imply each other: just assume one and prove the other, and then repeat the other way around. But to state that the two statements are in fact perfectly equivalent—there exists no universe in which one of these statements is true and the other false—seems like a much stronger thing. And yet it isn’t.

What other interesting equivalences (or alternative definitions) are there in mathematics?

Well, for example: the definition of a prime number.

Most people, if asked, will probably state that a number is prime if and only if it has no factors other than itself and $1$. This seems reasonable, and indeed it is the original motivating definition. However, we can also say that a number $n$ is prime if and only if whenever $n$ divides $ab$, it divides $a$ or $b$ (or both). The first implication (that this is true if $n$ is prime) is a classical theorem called Euclid’s lemma, but in fact the converse is true too and so the statements are equivalent. We might as well define prime numbers this way.

However, something interesting happens when we move from the integers to something more general. In a general ring, these apparently equivalent definitions separate into two distinct notions called ‘prime’ and ‘irreducible’. It just so happens that in the integers (and in general, in a certain class of rings called GCD domains) they coincide.

Another famous example of an interesting equivalence is of course due to the French mathematican Augustin-Louis Cauchy.

The standard definition of convergence of a real sequence is the following:

A sequence $(a_n): \mathbb{N} \to \mathbb{R}$ is said to converge (to $L \in \mathbb{R}$) if for all $\epsilon > 0$ there exists an $N \in \mathbb{N}$ such that for all $n > N$ we have $\lvert a_n – L \rvert < \epsilon$.

In words, it is saying that a sequence converges if and only if we can make the value of the sequence arbitrarily close (as close as we like) to the limit point $L$ by moving sufficiently far along the sequence towards infinity.

There are some functions that satisfy this definition for some (unique!) value of $L$, and some that don’t for any. Another similar but subtly different property a function might have is the following:

A sequence $(a_n): \mathbb{N} \to \mathbb{R}$ is said to be Cauchy if for all $\epsilon > 0$ there exists an $N \in \mathbb{N}$ such that for all $n, m > N$ we have $\lvert a_n – a_m \rvert < \epsilon$.

This one is saying, in words, that a sequence is Cauchy if and only if the terms in the sequence can be made arbitrarily close to each other by moving sufficiently far along the sequence.

As you may have suspected, these two properties turn out to imply each other: convergent sequences are Cauchy, and Cauchy sequences converge. The first implication is the easier one to prove, but both are true and therefore the two properties are equivalent. They are both perfectly good characterisations of convergence. This is particularly convenient since they contain an important conceptual difference: the second definition makes no mention of a limit, so we can prove convergence to some limit even when we don’t know what that limit actually is. Having these kind of equivalences is actually quite powerful!

So, what is the meaning of all of this? The question is perhaps not a mathematical one, and even then I’m not exactly sure of the answer.

My impression is this: definitions are less important than often assumed; it is the underlying properties of mathematical objects that are fundamental. Definitions are just a human way of putting an abstract concept into one-to-one correspondence with a more concrete ‘testable’ truth.

There are some sequences that are Cauchy, and some sequences that converge. In fact, we are talking about the same set of sequences that all have some ‘underlying property’ in common, but as humans we like to anchor this property to things we can more readily pin down in symbols—even if we end up doing so from two quite different angles.

## 466 Replies to “Abelianness and equivalent definitions in general”

1. Pingback: cialis site
2. Pingback: buy levitra
3. Pingback: i took viagra mom
4. Pingback: mail order viagra
5. Pingback: viagra discount
6. Pingback: viagra cost
7. Pingback: celebrex rating
8. Pingback: 20 mg buspirone
9. Pingback: is buspirone
10. Pingback: baclofen med
11. Pingback: atorvastatin s
12. Pingback: abilify definition
13. F*ckin’ remarkable things here. I’m very happy to look your article. Thank you a lot and i am taking a look forward to touch you. Will you please drop me a e-mail?

14. Hello, after reading this amazing post i am as well delighted to share my knowledge here with mates. Catlee Alleyn Bergerac

15. I enjoy what you guys tend to be up too. This kind of clever work and
exposure! Keep up the fantastic works guys I’ve incorporated
you guys to my own blogroll.

16. Hi there! Someone in my Myspace group shared this website
with us so I came to take a look. I’m definitely loving
the information. I’m book-marking and will be tweeting this to my followers!
Excellent blog and excellent design and style.

17. “I worked with your dad for almost 20 years and will miss him very much,” one colleague wrote. “He was so gracious… his enthusiasm for life was contagious… and his eyes lit up whenever he spoke of you and your brother.”
카지노사이트

18. “I want to tell you that your dad was always admirable,” a woman who went to university with him wrote to her in Spanish. “Very intelligent with a bright personality.”
메리트카지노

19. By then, his condition had become more serious, and he was eventually transferred to Houston Methodist Hospital. A few days later, he was placed on a ventilator.
예스카지노

20. “He was always saying, ‘Oh, I’m good. I survived another day,'” she said. “He was very protective of us about his condition. And I think when he went to the hospital, he had no idea that it would become so serious.”
메리트카지노

21. I’ve been surfing on-line more than 3 hours these days, yet I never discovered any fascinating article like yours.
It is lovely price enough for me. Personally, if all web owners
and bloggers made excellent content as you probably did, the web
shall be much more helpful than ever before.

22. Elias Cawthorn says:

YOU NEED FAST PROXY SERVERS ?

Check it out this Anonymous and Private Proxy Servers.
HTTP & SOCKS5 Proxy supported.
IP Authentication or Password Authentication available.
HERE: https://bit.ly/3ifZkmL

23. Hello! I’m at work browsing your blog from my new iphone!
Just wanted to say I love reading through your blog and look forward to all your posts!

Carry on the great work!

24. Pingback: sildenafil prices
25. I am sure this piece of writing has touched all the
internet viewers, its really really pleasant article on building up new blog.

26. Pingback: viagra on sale
27. I have been surfing on-line greater than 3 hours today, yet I by no means found any fascinating article like yours.

It is pretty worth enough for me. In my view, if all web owners and
bloggers made excellent content material as you
did, the internet will probably be much more helpful than ever before.

28. Pingback: viagra meme
29. Melina Woollard says:

YOU NEED QUALITY VISITORS FOR YOUR: xanderlewis.co.uk

WE PROVIDE ORGANIC VISITORS BY KEYWORD FROM SEARCH ENGINES OR SOCIAL MEDIA

YOU GET HIGH-QUALITY VISITORS
– visitors from search engines
– visitors from social media
– visitors from any country you want

CLAIM YOUR 24 HOURS FREE TEST => https://bit.ly/2HQZggh

30. Nathan Olsen says:

YOU NEED QUALITY VISITORS FOR YOUR: xanderlewis.co.uk

WE PROVIDE ORGANIC VISITORS BY KEYWORD FROM SEARCH ENGINES OR SOCIAL MEDIA

YOU GET HIGH-QUALITY VISITORS
– visitors from search engines
– visitors from social media
– visitors from any country you want

CLAIM YOUR 24 HOURS FREE TEST => https://bit.ly/2HQZggh

31. I needed to thank you for this wonderful read!! I certainly loved every little
bit of it. I have got you bookmarked to check out
new stuff you

32. Pingback: avanafil vs viagra
33. Greetings! Very useful advice in this particular article!
It’s the little changes that make the biggest changes.

Thanks for sharing!

34. Pingback: sildenafil
35. I am sure this post has touched all the internet
viewers, its really really nice post on building up new weblog.

36. It is perfect time to make some plans for the future and it is
time to be happy. I’ve read this post and if I could I desire to suggest you few interesting things or
tips. Perhaps you could write next articles referring to this article.
I want to read even more things about it!

37. Likes on Instagram can improve your brand’s engagement numbers. When you buy Instagram likes you get more engagement on your profile which means more reputation.

38. Likes on Instagram can improve your brand’s engagement numbers. When you buy Instagram likes you get more engagement on your profile which means more reputation.

39. Pingback: atorvastatin aka
40. Way cool! Some very valid points! I appreciate you writing this article plus the
rest of the site is also very good.

41. Pingback: viagra uk
42. Ahaa, its pleasant dialogue on the topic of this paragraph here at this webpage,
I have read all that, so now me also commenting at this place.

43. Pingback: marley drug viagra
44. The other day, while I was at work, my cousin stole my iphone and tested to see if
it can survive a thirty foot drop, just so she can be a
youtube sensation. My iPad is now destroyed and she has
83 views. I know this is totally off topic but I had to share it with someone!

45. Pingback: cialis or viagra
46. I am sure this paragraph has touched all the internet people,
its really really fastidious post on building up new website.

47. Pingback: cialis viagra
48. Likes on Instagram can improve your brand’s engagement numbers. When you buy Instagram likes you get more engagement on your profile which means more reputation.

49. I am sure this article has touched all the internet users,
its really really nice paragraph on building
up new webpage.

50. These are genuinely impressive ideas in concerning blogging.
You have touched some nice points here.
Any way keep up wrinting.

It’s the little changes that produce the most significant changes.

Many thanks for sharing!

52. Pingback: viagra working
53. I am sure this piece of writing has touched all the internet users,
its really really nice post on building up new web site.

54. Pingback: viagra 100mg
55. I will immediately grasp your rss feed as I can’t to find your email subscription hyperlink
or newsletter service. Do you’ve any? Please permit me know so that I may just subscribe.
Thanks.

56. Pingback: otc viagra
57. I enjoy whst yoou guyus end tto bbe upp too. Suchh clewver woork andd reporting!
Keeep upp the great woirks guyhs I’ve included yoou gguys too mmy peersonal blogroll.

58. Pingback: cialis copay card
59. Greetings from Idaho! I’m bored to death at work so I decided to browse your
blog on my iphone during lunch break. I enjoy the information you present
here and can’t wait to take a look when I get home. I’m surprised at how fast
your blog loaded on my mobile .. I’m not even using WIFI, just 3G ..
Anyhow, amazing site!

60. I’ll right away snatch your rss feed as I can’t to find your e-mail subscription link or newsletter service.
Do you have any? Kindly permit me recognize so that I may subscribe.
Thanks.

61. As I web site possessor I believe the content matter here is rattling excellent , appreciate it for your efforts. You should keep it up forever! Best of luck.

62. Nadine Kittredge says:

YOU NEED QUALITY VISITORS for your: xanderlewis.co.uk

My name is Nadine Kittredge, and I’m a Web Traffic Specialist. I can get:
– visitors from search engines
– visitors from social media
– visitors from any country you want
– very low bounce rate & long visit duration

CLAIM YOUR 24 HOURS FREE TEST => https://bit.ly/3nnmM4I

63. Hi my family member! I wish to say that this post is awesome, nice written and come with almost all vital infos. I would like to peer more posts like this .

64. Great V I should certainly pronounce, impressed with your website. I had no trouble navigating through all the tabs and related info ended up being truly easy to do to access. I recently found what I hoped for before you know it in the least. Quite unusual. Is likely to appreciate it for those who add forums or anything, web site theme . a tones way for your customer to communicate. Nice task..