## Thursday, February 21, 2008

### In light of my last post

My abstract algebra midterm had a delightful question on it that was just too easy:
If G is a group in which every element is its own inverse, then G is abelian.
(HA! Too easy. I LOVE proving that things are abelian, and this is super easy. Wanna see? Too late!)

NOTE: I am not saying that if you can't do this you are not smart, but compared to other things in group theory I think this is one of the easiest things to do.

NTS. If G is a group in which every element is its own inverse, and $a^2b^2=(ab)^2$ then G is abelian.
Proof:
$a, b\in G, a^{-1}=a$ and $b^{-1}=b$
$a^{-1}=a$
$aa^{-1}=aa$
$e=aa$
We can do the same for b.

$a^2b^2=(ab)^2$
$aabb=abab$
$ee=abab$
$(aee=ae=a)$
$a=aabab$
$ab=aababb$
Since $aa=e=bb$ we can cancel and therefore $ab=ebae=ba$ which shows that G is abelian.
QED.

I also finally just got latex for blogger to work and wanted to use it. Also, I just really liked doing this proof. It's fun, and did I mention easy?

#### 13 comments:

Terry said...

Math joke alert! Check this out:
http://www.irregularwebcomic.net/1640.html

T.O. said...

Funny coincidence: I was doing this exact problem today! It finally occurred to me, after staring at your post in confusion for a while, why we can say a^2b^2 = (ab)^2. Thanks!! Who knew feminism would help me with my math homework?!

lost clown said...

left and right cancellation is a wonderful thing. I love proving things are abelian, it's so straightforward.

Well, this is the blog of a feminist mathematician/physicist. Glad I could help!

Dr. Brazen Hussy said...

Dude, I have no idea what any of that meant. You are, like, so much smarter than me.

lost clown said...

Yes, because anyone can type and sequence DNA (and do all that other stuff you do that I don't quite understand).

Dr. Brazen Hussy said...

DNA sequencing is easy! All you need is an opposable thumb. I'm planning to train a monkey to do my work for me... just as soon as I can find a monkey.

lost clown said...

I want a monkey!

(and what about all that other stuff? hmmmm?)

Dr. Brazen Hussy said...

I don't quite understand all that other stuff either. :)

Maggie Hays said...

I'm not looking forward to being given formulae like these to work on in my math class this semester! (despite the fact that I passed my final math exam in high school years ago -- it's been a long time)

For sure there will be a lot of revision and homework to do!

lost clown said...

You're studying abstract algebra?

Maggie Hays said...

Not at the moment. (*sigh*) At the beginning of my semester, I'm mainly required to work out numeracy and arithmetic exercises. But, I know there will be a re-introduction to algebra at some point (as the teacher mentioned to us). I don't know about abstract algebra. It's been years since I left high school (and BTW there were very hard equations at the final math exam as well as some formulae that looked like the ones in your post - I remember). So, what's the difference between abstract algebra and any other kind of algebra, Lost Clown? What are the other kinds of algebra? Enlighten me!

ilovenumb3rs said...

So, I haven't taken a decent math course in years, mostly because I haven't had to. The funny thing is that I get it. I followed along with your proof with no problem. Does that make me a genius. lol. I miss math.

lost clown said...

No, it just means you like Algebra. ;)