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        riddles >> putnam exam (pure math) >> Multiplication of a Rotation and a Reflection
        
(Message started by: Whiskey Tango Foxtrot on Nov 30^th, 2006, 9:32am)

Title: Multiplication of a Rotation and a Reflection
Post by Whiskey Tango Foxtrot on Nov 30^th, 2006, 9:32am

The multiplication of a rotation, R, about the z-axis and through the angle pi and a reflection in the xy-plane yields a group. Show this group to have four elements, then write it as a direct product.

Title: Re: Multiplication of a Rotation and a Reflection
Post by Icarus on Nov 30^th, 2006, 7:52pm

The first, A, carries (x,y,z) to (-x, -y, z). The second, B, carries (x,y,z) to (x, y, -z). Clearly A² = I, B² = I, and AB = BA. Hence |<A>| = 2, |<B>| = 2, and the full group is <A> x <B>, and |<A> x <B>| = 4.

I suppose you could say that I didn't solve it, since I showed the direct product first, and not the 4 elements... ::)

Title: Re: Multiplication of a Rotation and a Reflection
Post by Whiskey Tango Foxtrot on Nov 30^th, 2006, 10:38pm

It's fine by me and I'm the one who posted the riddle. I guess that means you win. It's nice to see some other people are interested in Group Theory. Now that I've been studying it, I don't understand why I wasn't taught it earlier. It should be right up there with algebra with respect to its universal applications. At least that's how I see it.

Title: Re: Multiplication of a Rotation and a Reflection
Post by SMQ on Dec 1^st, 2006, 5:44am

on 11/30/06 at 22:38:18, Whiskey Tango Foxtrot wrote:

I guess that means you win. It's nice to see some other people are interested in Group Theory.

I'm interested, but despite my undergrad math minor I never learned any in school, and since it has exactly zero applicability to my actual for-pay job writing real estate software, I have to pick it up on my own time... any recommendations for good introductory material -- preferably online?

--SMQ

Title: Re: Multiplication of a Rotation and a Reflection
Post by Whiskey Tango Foxtrot on Dec 1^st, 2006, 8:51am

http://www.math.niu.edu/~rusin/known-math/index/20-XX.html
A decent introduction to the ideas behind Group Theory.

http://web.usna.navy.mil/~wdj/tonybook/gpthry/node1.html
Pretty deep and well thought out. The pages are a little too dense for me, but it is still manageable.

http://www.nbi.dk/GroupTheory/
Not too great but it's free and it covers some of the commonly used applications.

If you really want to understand Group Theory, pick up an old textbook. They're relatively cheap. If you have any kind of interest in physics, I can highly recommend "Symmetry in Physics" by Elliott and Dawber. If you still have access to a college library I'm sure they have several very good books.

Title: Re: Multiplication of a Rotation and a Reflection
Post by Barukh on Dec 1^st, 2006, 9:01am

on 12/01/06 at 05:44:42, SMQ wrote:

any recommendations for good introductory material -- preferably online?--SMQ

I would suggest you find the following books:

Herstein "Topcis on Algebra" (ch. 2)
Fraleigh "A First Course in Abstract Algebra" (ch. 1-2).

For the geometric interpretation of groups, Coxeter's "Regular Polytopes" is an excellent source.

Title: Re: Multiplication of a Rotation and a Reflection
Post by Whiskey Tango Foxtrot on Dec 1^st, 2006, 9:10am

Herstein can be hard to find. My friend came across a used copy a while ago. I read portions of it, most of chapters 1 and 2 I think, and thoroughly enjoyed it.

Title: Re: Multiplication of a Rotation and a Reflection
Post by Barukh on Dec 1^st, 2006, 10:03am

To the question of WTF: Consider the following degenerated polygon: it has 2 vertices, connected by two distinct (curved) sides (it is called 2-gon). Put it in the 3D-space so that its vertices lie on z-axis. Then, operation A interchanges only sides, operation B interchanges only vertices, and AB = BA interchange both vertices and sides. Thus, we get the full symmetry group of a 2-gon, which is of course dihedral group D₂ = C₂ x C₂.

Title: Re: Multiplication of a Rotation and a Reflection
Post by Sameer on Dec 9^th, 2006, 2:04am

on 12/01/06 at 09:10:08, Whiskey Tango Foxtrot wrote:

Herstein can be hard to find. My friend came across a used copy a while ago. I read portions of it, most of chapters 1 and 2 I think, and thoroughly enjoyed it.

You can find it on amazon ... btw i do remember this book because earlier I had asked the same question and Icarus suggested this book... I got it and it is pretty awesome.. even though mathematics is not my primary field (am an EE ;)) i do really like math and this is one good book... Farleigh is expensive... i think I saw it for 150 bucks.. thats expensive for my pass reading :-[

Title: Re: Multiplication of a Rotation and a Reflection
Post by Icarus on Dec 9^th, 2006, 7:03am

on 12/09/06 at 02:04:28, Sameer wrote:

btw i do remember this book because earlier I had asked the same question and Icarus suggested this book...

Not me. I've never read it.

Title: Re: Multiplication of a Rotation and a Reflection
Post by Sameer on Dec 9^th, 2006, 12:11pm

on 12/09/06 at 07:03:58, Icarus wrote:

Not me. I've never read it.

Whoops :-[ it was a long time ago.. but somebody definitely suggested it.. so thank you whoever did!! :)

Edit: Apparently it was Barukh the same ol' person who suggested it :P .. i found the post where I asked the question and got the response

http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_easy;action=display;num=1078017611;start=

Icarus you did suggest the book on number theory though, which is indeed awesome...

Of course I read through the post and saw my newbie ramblings in there without contributing anything to the posted problem
:-[

Title: Re: Multiplication of a Rotation and a Reflection
Post by THUDandBLUNDER on Dec 9^th, 2006, 12:22pm

Barukh recommended it again (to SMQ) in this thread only 8 days ago.

I have quite a few Algebra ebooks that I am willing to share if anybody is interested.

Title: Re: Multiplication of a Rotation and a Reflection
Post by Whiskey Tango Foxtrot on Dec 10^th, 2006, 11:55am

I'd love an algebra ebook. :)

Title: Re: Multiplication of a Rotation and a Reflection
Post by Sameer on Dec 10^th, 2006, 12:57pm

Yes me too...