Question about Class equation?

[colum]

In Algebra (Author : M.Artin )

In Chapter 5, He writes about Abstract group action.
And in Chapter 6, He writes about Conjugacy class and Class equation

I think there is a little different thing between orbit and conjugacy class.
Only thing is that orbit is more general thing.
When we choose operation as conjugation, we call this orbit as conjugacy class.
Am I right?

So, I'm very confusing about that.
I don't know why we use group action.
just for classify group?
And I don't know when I use operation as left multiplication or conjugation.
So, do you mind if I ask you an example of that?

I see an example about group action.
It is classify cube... but I don't know how I can use it on general group.


And one more question.
This is my thought of classifying group.

When a group is given, we can easily know about its order.
[ In many problem, they suggest just information about order.. ]
So we use Sylow theorem to know about Sylow p-subgroup.
And we combine that group by using semidirect product.
In good case, we just use direct product.
Is there anything wrong about that?

[adalwine]

In Algebra (Author : M.Artin )

In Chapter 5, He writes about Abstract group action.
And in Chapter 6, He writes about Conjugacy class and Class equation

I think there is a little different thing between orbit and conjugacy class.
Only thing is that orbit is more general thing.
When we choose operation as conjugation, we call this orbit as conjugacy class.
Am I right?

So, I'm very confusing about that.
I don't know why we use group action.
just for classify group?
And I don't know when I use operation as left multiplication or conjugation.
So, do you mind if I ask you an example of that?

I see an example about group action.
It is classify cube... but I don't know how I can use it on general group.


And one more question.
This is my thought of classifying group.

When a group is given, we can easily know about its order.
[ In many problem, they suggest just information about order.. ]
So we use Sylow theorem to know about Sylow p-subgroup.
And we combine that group by using semidirect product.
In good case, we just use direct product.
Is there anything wrong about that?

there are different possible ways a group G might act on a given set X.

basically for a finite set, there are "two extremes":

G acts on X faithfully as a subgroup of Sym(X) (this can always happen when |G| ? |X|,

but may not be possible if |G| > |X|).

G acts on X trivially as the identity function.

a nice example of a group acting on a set is the dihedral group of order 2n

which acts on the regular n-gon in the plane. we can imagine the n-gon

as consisting of either n points in the plane (the vertices), or n line segements in the plane (the edges), which the rotations and reflections permute. the explicit homomorphism from Dn to Sym(X) is then:

r ? (1 2.....n) (an n-cycle of the edges/vertices)
s ? (2 n)(3 n-1).....((n/2) (n/2 + 1)) if n is even
...? (2 n)(3 n-1).....((n+1)/2 (n+3)/2) if n is odd

of course we can always take X = G (that is, let G act on itself).

if the action is conjugation, the orbits are the conjugacy classes.
if the action is left-multiplication, we only get one orbit, since

h = (hg^-1)g, so any two elements g,h lie in the same orbit.

(an action which only has one orbit is called transitive).

since left-multiplication is a (faithful) action, we get that G

is isomorphic to a subgroup of Sym(G) ? S_|G|

if |G| is finite. this is what Cayley's theorem states.

for example, the cyclic group C4 is isomorphic to a subgroup of S4,

namely: {e, (1 2 3 4), (1 3)(2 4), (1 4 3 2)}.

it often turns out that G acts faithfully on a finite set X (like a cube, with a finite set of vertices), and if we regard G as a subgroup of Sym(X), then G?Alt(X) (where Alt(X) is the alternating subgroup) is the set of "orientation-preserving symmetries".

a third important action is the induced action on a coset space G/H, for a subgroup H, given by:

g.(xH) = (gx)H

this is an action because:

g.(g'.xH) = g.((g'x)H) = g(g'x)H = (gg')xH = (gg').xH and

e.(xH) = (ex)H = xH

this tells us we have a homomorphism of G into Sn, where n = [G:H], which can tell us something about which sizes of normal subgroups may be possible.

group actions allows us to use symmetry groups, to study symmetric OBJECTS. the cube itself has no algebraic structure, but its symmetry group does. one application of this is in tiling: many mosques in the middle east are an example of how symmetry can be used to create interesting decorative patterns (one looks for small "base patterns" that can be translated, rotated and mirrored to fill a surface).

in other words: if a group can act on a set, then the set gains a kind of "group structure", and we can use this to find out things about the set. let me make a sort of analogy:

with a vector space, we have two structures: an abelian group V, and a field F. what turns the abelian group V into a vector space V is the scalar multiplication:

a.v = av

2 of the things we require is that this be a group action of F* on the set V:

a.(b.v) = a.(bv) = a(bv) = (ab)v = (ab).v
1.v = 1v = v

of course V is also an abelian group, so we further require that the action is a +-homomorphism:

a.(u+v) = a(u+v) = au+av = a.u + a.v

and (F,+) is an abelian group, so we require that the action is a homomorphism from (F,+) to (V,+)

(a+b).v = (a+b)v = av + bv = a.v + b.v

so a group action of a group on a set is just like "scalar multiplication", but we forget anything having to do with addition. this turns X into a new kind of thing, a G-set, which is sort of "half set and half group" (just like a vector space is "half abelian group and half field"). G-sets aren't groups, but some of the properties of the group G make themselves felt in the G-set. one example of this is when we split G into conjugacy classes, the order of G limits the sizes we can have for conjugacy classes, because each one has to have size a divisor of the order of G.

as far as semi-direct products go, yes, you're on the right track. however, a given group G may not have any normal sylow p-subgroups, for any prime p dividing |G|, and you need a normal subgroup to build an internal semi-direct product.

an interesting example is Q = {1,-1,i,-i,j,-j,k,-k} the quaternion units group. this is a p-group (with p = 2), and it cannot be realized as a semi-direct product, even though it is not simple.