Group Theory: Necessary & Sufficient Conditions for G_W = G_(W)

In summary, the conditions necessary and sufficient for G_W to equal G_(W) are that: 1) G_W is a subset of G; and 2) for every w in W, g(w) = w for all w in G_W.
  • #1
mehrts
15
0
Let n be in |N. Let G denote S_n , the symmetric group on n
symbols. Let W be a subset of {1, 2, ..., n}.

Write down VERY simple
necessary and sufficient conditions on |W|,

for G_W to equal G_(W).
We know G_W < G_(W) < G , but now what ?
 
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  • #2
Care to define GW and G(W)?
 
  • #4
Okay, this is a very simple problem. Have you tried anything?
 
  • #5
We just started the topic and all I need is a hint on how to start the problem. :(
 
  • #6
That jpeg is far too small to read for me. What are G_W and G_(W)? The hint is, as ever, start by writing out the definitions and what you want to prove.
 
  • #7
Click on the jpeg to see a bigger picture.

http://img226.imageshack.us/my.php?image=untitled1nx0.jpg
 
  • #8
G_W ={ g in G : g(w)=w for all w in W}

G_(W) = { g in G : g(W)=W}why not just type it? The first is the subgroup of G that fixes W elementwise (i.e. fixes every element of W), and the second the subgroup that fixes W setwise (i.e. permutes the elements of W amongst themselves), so they're different precisely when there is something in G(W) that is not in G_W. So, writing out what the definition means that the answer is...
 
  • #9
Say n = 4, W = {1,4}. What are GW and G(W)?
 
  • #10
For example,

Let S = {1, 2, 3, 4}.

If W = {1}

G_W = G_(W) = {(1),(2 3 4),(2 4 3),(3 4),(4 2),(2 3)}.

IF W = {1, 2}

G_W = {(1), (3 4)}

G_(W) = {(1), (1 2), (3 4), (1 2)(3 4)}

If W = {1, 2, 3}

G_W = {(1)}.

G_(W) = {(1), (2 3), (3 1), (1 2), (1 2 3),(1 3 2)}.

So can I conclude that |W|= 1 if G_W = G_(W) ?
 
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  • #11
From this alone, no you can't conclude that. Ultimately, you can, but you have to provide a real proof. Also note that you are asked to find a necessary and sufficient condition for GW = G(W). This means you need to conclude not only that |W| = 1 if GW = G(W), but also that GW = G(W) if |W| = 1.
 
  • #12
Thanks.
So their is only one necessary and sufficient condition then ?
Yup, the second part of the question was asking to prove the conditions are necessary and sufficient. :)
 
  • #13
I think the correct answer would be that |W| = 0 or 1. Since the empty set contains the identity mapping. Is this correct ?
 
  • #14
You're right, |W| = 0 or 1. The empty set does not contain the identity mapping, the empty set contains nothing, that's why it's called the empty set. So prove that if |W| = 0 or 1, then GW = G(W), and also prove the converse, i.e. if GW = G(W), then |W| = 0 or 1.
 
  • #15
Thanks alot. :)
 

FAQ: Group Theory: Necessary & Sufficient Conditions for G_W = G_(W)

What is group theory?

Group theory is a branch of mathematics that studies the properties of groups, which are sets of elements that follow certain rules or axioms. Groups are used to describe the symmetries and transformations of mathematical objects, and have applications in many areas such as physics, chemistry, and computer science.

What are the necessary and sufficient conditions for G_W = G_(W)?

The necessary and sufficient conditions for G_W = G_(W) are that the group G must act on the set W, and that every element of the group must have a unique image in the set W. This means that for every g in G, there must exist a unique w in W such that g(w) = w. In other words, every element of the group must map to a different element in the set W.

What does G_W = G_(W) represent?

G_W = G_(W) represents the stabilizer subgroup of a group G with respect to a set W. This means that it is the subgroup of G that leaves every element in the set W unchanged when it acts on it. In other words, the elements of G_W are the ones that do not change the elements of W when they are applied to them.

How is group theory used in science?

Group theory has many applications in science, including in physics, chemistry, and computer science. In physics, it is used to study the symmetries of physical systems and to understand the fundamental forces of nature. In chemistry, it is used to describe the symmetries and electronic structures of molecules. In computer science, it is used to study algorithms and data structures.

What are some examples of groups in science?

There are many examples of groups in science, including the symmetric group, which describes the symmetries of geometric figures, and the rotation group, which describes the rotations of an object in three-dimensional space. Other examples include the permutation group, the dihedral group, and the special orthogonal group. These groups have applications in various fields such as crystallography, quantum mechanics, and molecular biology.

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