/sci/ - Science & Math

[Anonymous No.2929613]

tell me all you know about group theory

[Anonymous No.2929631]

when you get ppl into a group
eventually someone farts

[Anonymous No.2929632]

>>2929627
it does not look like i will be told much...

[Anonymous No.2929627]

>>2929613

bumping out of interest in what you will be told.

[Anonymous No.2929634]

>>2929613
>>2929627
>>2929632

samefags

http://en.wikipedia.org/wiki/Group_theory

[Anonymous No.2929636]

>>2929634

actually not, and yes we all know about wikipedia. looking for some more interesting insights from those in the know

[Anonymous No.2929637]

>>2929613
what do you wanna know?
You know the definition, what are your open questions?
I mean "tell me all you know" is a bit vague.
We can chat about Lie Groups, I like em :)

[Anonymous No.2929641]

>>2929637

any interesting properties that have interesting implications/uses, ways of visualising some of the basic concepts, anything you like really

[None !nH/8MpZld2 No.2929648]

>>2929634

No. lrn2samefagdetect

[Anonymous No.2929650]

"Hey guys, tell me everything you know about an entire branch of mathematics"

[Anonymous No.2929659]

>>2929641
anything is too vague.
you know it's a fucking huge topic as one would guess from this page
http://en.wikipedia.org/wiki/List_of_group_theory_topics

okay random fact
<span class="math">Sp(1) \cong SO(4)/SO(3) \cong SU(2) \cong Spin(3)[/spoiler]
these are all well known Lie Groups and they turn out to be the same. "SU(2)" is probably the most commonly used name for it.
Moreover, topologically this group is equal to S^3, the threedimensional Sphere. It can easily be made to be a Riemannian manifold (this is also "obvious from the fact that the group is compact").
Moreover, there is a nice construction of this sphere
http://en.wikipedia.org/wiki/Hopf_fibration
The group is also the symmetry groups of spin 1/2 fermions (electrons).
Note that "all compact groups" (not technically) can be represented unitary (they leave some Hilbert Space product/expectation value (in Quantum Theory) invariant)

[Anonymous No.2929660]

>>2929650
just if people wanna drop in and muse on what they think are some interesting things in group theory

[Anonymous No.2929663]

addon: There are actually two of those links

http://en.wikipedia.org/wiki/List_of_group_theory_topics
http://en.wikipedia.org/wiki/List_of_Lie_group_topics

[Anonymous No.2929672]

>>2929659

i'm unaware of what lie groups even are.. care to give a laymans on them? i'm not being lazy and not going to wikipedia, i think i'd have a better understanding if explained to me

[Anonymous No.2929692]

>>2929672
Lie Groups are (smooth) manifolds (surfices) where it makes sense to multiply points.

It's like if you would be able to multiply the end of your thumb T with the tip of your nose N and there would be an actual rule where you find the point A=T·N on your body.

An example is "all real numbers grater than zero".

T=exp(3)=20.085... is such a number and N=exp(1.4)=4.055... is one as well.
And so is
A=T·N=exp(4.4)=81.4509...
You can multiply points of a smoothly connected space.

another example is the rotation around an axis. the manifold is a circle and if you rotated by an angle a and then by an angle b, then you've rotated by an angle a·b.

Read my previous post again. It says that the 3-Sphere, that is the set of points given by
x^2+y^2+z^2+w^2=1
has a group strutuce - and this is even a very very relevant group. (in qm it explains why matter is stable)

However I'm not going to give you random facts anymore, that seems like a "wast of time".

[Anonymous No.2929699]

>then you've rotated by an angle a·b
sorry, here I used angles as synonymos with rotations.
what you multiply using "·" are the rotations, of course the angles just add up with "+".

[Anonymous No.2929703]

>>2929692

why does your real numbers example not extend to all real numbers (not just greater than 0)?

Is it because 0 has no inverse to take it to the identity? would it apply to R \ {0}

[Anonymous No.2929723]

>>2929703
hehe, good remark :)
(are you OP?)
R \ {0} is, in fact, also a Lie Group, however a more complicated one.
The point is that this group, as a manifold, is the Line R without one point, the zero, which therefore cuts the group into two pieces. The group isn't connected anymore (there are three math wiki articles on the topic of "connectedness" as I just saw) and you really want your group to be simply connected
http://en.wikipedia.org/wiki/Simply_connected
I mean your group is nice, but not so easy anymore to study.
See, you want a Lie Group element A to be represented by A=exp(a)
http://en.wikipedia.org/wiki/Exponential_map
(note that this is a vast generalisation of the exponential function)
and if you take a from the reals, then exp(a) is always positive.
this is where my example came from ;)
the set of elements a, i.e. those which are argument of exp, are the elements of the Lie Algebra
http://en.wikipedia.org/wiki/Lie_Algebra
which in group theory is (most of the times) defined to be the tangent space of a lie group (at one special point, the unit element)

btw. your group R\{0}
is just my group cross Z_2=Z/2Z, i.e. the product of a Lie Group and the smallest abelian group.
there is even a song about this small group:

http://www.youtube.com/watch?v=BipvGD-LCjU

(ONE OF THE NERDIEST SONGS I KNOW!)

[Anonymous No.2929760]

>>2929723

ah i okay i see, gonna need to read up on a few things like connectedness.

quickk question what is the group Z_2=Z/2Z ?
that's not a quotient group is it?

p.s that song's awesome, you heard this?
http://www.youtube.com/watch?v=ES-yKOYaXq0

[Anonymous No.2929799]

>that's not a quotient group is it?
yes it is
http://en.wikipedia.org/wiki/Cyclic_group
Z/nZ is
the integers with "+" where a+ni=a for example
Z/5Z is "the circle
0123401234012340123401234
hence 2+4='6'=1
you always factor out 5Z.

Z/Z2 (the finite simple group of order two *sing*)
is the group
0101010101010 with puls
i.e.
0+0=0
0+1=1
1+0=1
1+1=0
this group is abelian (because 0+1=1+0) and therfore you might use "+" as the operaton. however, in our context it's better to use a representation with "·" i.e.
you identify

+0 with ·1 (the neutral elements, respectively)
and
+1 with ·(-1)

i.e.

1·1=1
1·(-1)=(-1)
(-1)·1=(-1)
(-1)·(-1)=1

notice that this is the same structure as above just with · and not +. They are isomorphic, see also
http://en.wikipedia.org/wiki/Representation_theory

now my group was {A|A=exp(a), a element reals}, i.e. all the numbers R>0
and your group was R\{0}
but
(R\{0})=+/- exp(a)=+/- (R>0)
so
your group = my group cross Z/2Z
;)

[Anonymous No.2929855]

in what I said above read
>>2929659
again.

>Sp(1) \cong SO(4)/SO(3) \cong SU(2) \cong Spin(3)
these are all well known Lie Groups and they turn out to be the same.
>"SU(2)" is probably the most commonly used name for it.
>Moreover, topologically this group is equal to S^3, the threedimensional Sphere.

i.e.
S^3=SO(4)/SO(3)
as a manifold.
this says that if you take all the rotatons SO(4) in a 4dimensional (flat) space and view itself as a space, i.e. "the space of all the angles-directions in 4 dimensions you could possibly go" (thats vague) and then factor out SO(3) - you "identify all 3-dimensional directions to be the same" - then what is left is a space with 3 dimensions which is the S^3, the spere in 3 dimesnions.

how do the dimensions work out?
well in 4-dimensional space, lets call is x,y,z,t, there are not 4 angles for SO(4) but 6!
x,y
x,z
y,z
t,x
t,y
t,z
and therefore you cann easily kill three of the directions which are identified via SO(3)
therefore
SO(4)/SO(3) is still 3-dimensional ^^

and the statements says that it's also a lie group - it's not rotations, like SO(3), but something else.

Well, the other equality sign says it already
SO(4)/SO(3)=Spin(3)
it's the 3-diemsnional spin group - Quantum Mechanics here we come.

http://en.wikipedia.org/wiki/Special_orthogonal_group
http://en.wikipedia.org/wiki/Spin_group

The spin groups are in some sense bigger than the SO groups. they are the universal covers

http://en.wikipedia.org/wiki/Universal_cover

[Anonymous No.2929904]

k, I'll grab something to eat - if you're still on this topic and come up with another question i'll answer is in about 2 hours

[Anonymous No.2930297]

re