Another question from Elements of Abstract Algebra by Clark - transformation groups

In summary, given a group G of transformations that acts on X and a subgroup Go of G, where g * x = x for all x for each g in Go, the quotient group G/Go can act effectively on X if the G-action of elements in Go is known. This can be exploited to show that the G/G0 action is well defined by showing that if [h1] = [h2], then h1 * x = h2 * x for all x in X. In other words, the quotient group G/Go acts effectively on X by only considering the G-action of elements in Go.
  • #1
learningphysics
Homework Helper
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This is question 53[tex]\gamma[/tex]. Given a group G of transformations that acts on X... and a subgroup of G, Go (g * x = x for all x for each g in Go), show that the quotient group G/Go acts effectively on X.

A group G "acts effectively" on X, if g * x = x for all x implies that g = e, where g is a member of G.

I don't see how the quotient group G/Go can act on X... Each member of the quotient group, is itself a set of transformations. For example, take Go which is a member of G/Go. It seems to me that Go * x (where x belongs to X) is undefined, since Go is not a one to one correspondence from X to X (each member of Go is, but Go itself isn't).

I'd appreciate any help. Thanks.
 
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  • #2
Hint.

Let h be in G, so [h] = hG0 is in G/G0. Is [h]*x := h*x well defined?
 
  • #3
George Jones said:
Hint.

Let h be in G, so [h] = hG0 is in G/G0. Is [h]*x := h*x well defined?

Hi George... Thanks for the reply.

It seems to me like it is not well defined. There may be two different functions... say h1 and h2, such that [h1] =

... So is [h1]*x = h1*x or h2*x ?

 
  • #4
learningphysics said:
So is [h1]*x = h1*x or h2*x ?

In general, yes. However, in this case, we know something about the G-action of elements of G0. Can this be exploited to show that G/G0 action that I gave is well defined?
 
  • #5
You should try to verify this yourself, it is quite straightfoward. What does [h1]=

mean? That there is a k in Go such that h1k=h2. What was the definition of Go?

 
  • #6
Ah... I see now... h1*x = h2*x. Thanks George and Matt.
 
Last edited:
  • #7
learningphysics said:
Ah... I see now... h1*x = h2*x.

Careful - this isn't isn't necessarily true. But what is true?

Maybe you just made a typo.
 
  • #8
George Jones said:
Careful - this isn't isn't necessarily true. But what is true?

Maybe you just made a typo.

Hmm... If [h1]=

then there's a k in Go such that h1= h2k so

h1 * x = h2k * x
h1 * x = h2 * (k * x), then since k is in Go, k * x = x

h1 * x = h2 * x

I'm probably making a really stupid mistake somewhere. Sorry guys... I appreciate the patience.

 
  • #9
Sorry - my mistake.

Edit: I was thinking of the end result, i.e, the G/Go action.
 
  • #10
if you call all the elements of G that act trivially, "the identity", then the only elements that act trivially after that are called the identity.
 

1. What is a transformation group?

A transformation group is a set of transformations that can be applied to an object or space to create new versions of it. These transformations follow certain rules and can be combined together to create even more transformations.

2. What are the elements of a transformation group?

The elements of a transformation group are the individual transformations that make up the group. These transformations can include rotations, reflections, translations, and any other type of geometric transformation.

3. How is a transformation group represented?

A transformation group can be represented in different ways, depending on the context. In abstract algebra, it is often represented as a set of transformations with a binary operation that combines two transformations to create a new one. In geometry, it can be represented as a set of matrices or equations that describe the transformations.

4. What is the importance of transformation groups in abstract algebra?

Transformation groups are important in abstract algebra because they provide a way to study and understand symmetry and structure in mathematical objects. They also have applications in other areas of mathematics, such as group theory and geometry.

5. How are transformation groups used in real life?

Transformation groups have many practical applications in real life, such as in computer graphics, robotics, and physics. They are used to describe the movement and behavior of objects in space, and can also be used to create visual effects and animations.

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