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# Normalisers of a group. watch

1. I am completely lost with this question and any input or help or even a point in the right direction would be great; (sorry I dont know how to put in correct symbols, etc)

Let G be a group and a e G
The normaliser of a in G is Na(G)={g e G :ga=ag}

Find the normalisers of (1)(23) and (123) in S3.

By my interpretation of this, the only thing I can gather is ga=ag may mean that G is abelian or commutative? I have a feeling it has something to do with cosets however I'm unsure how to work out the coset in this situation, especially with permutation groups and I find numbers appearing and I dont know where they've come from. If you were working out cosets would you use (1)(23) and (123) as a subgroup so {(1)(23), (123)}

Any clarity on this question would be a great help because I'm majorly confused and dont even know if i'm coming at it from the right angle.

Thanks
2. It seems weird that you're talking about the normaliser of an element; usually normalisers refer to subgroups and centralisers refer to elements. And also, that notation's a bit weird; I'd say it's more common to see (or if you must) rather than . If you've got these from your lecture notes then it might be worth asking your lecturer about this point, because I'm not convinced it's even correct.

So for the sake of my explanation I'll talk about centralisers, where . That is, the centraliser of an element is the set of elements of the group which commute with that element.

Now, if then it doesn't mean that G is abelian. G is abelian if for all , but here we're just talking about the specific element a. Also, we only have that when , although it does mean that if then the element commutes with every element in G. However, even in this case it wouldn't mean it's an abelian group. For example, consider the group of 2x2 matrices (with nonzero determinant). Then commutes with every element (so the centraliser is the whole group), and yet the group is non-abelian.

Going on to your specific problem, to find the centraliser of an element in a permutation group, it's a good idea to consider conjugation. Note that . What happens to a permutation when you conjugate it? (Hint: see my reply to the recent thread on conjugacy classes about what happens when you conjugate a permutation; I'm too lazy to type it out again.)
3. I've literally copied the notation straight off a coursework question, it's confusing me because when looking on the internet all the notation is the same as you have written above so i'm not entirely sure what I'm looking for in the first place or how to do it!

If you are finding a centre of a group is that the same as a centraliser? because I have another question which is about that, which I also really dont understand so the information above that you've given me may be of better use on that instead?? However the notation I've got for that one at the start is

"The centre of G is Z(G)= {g e G : ga=ag, for all a e G" which isn't the same as above either

I'm really frustrated because I literally dont have a clue so cant even attempt it
4. (Original post by belle_2106)
I've literally copied the notation straight off a coursework question, it's confusing me because when looking on the internet all the notation is the same as you have written above so i'm not entirely sure what I'm looking for in the first place or how to do it!
It's definitely worth emailing someone to find out what's happened; I'm fairly sure it's non-standard to talk about the normaliser of an element. If you're confused by what I'm saying, then just replace "centraliser" by "normaliser" and by (or or whatever).

(Original post by belle_2106)
If you are finding a centre of a group is that the same as a centraliser? because I have another question which is about that, which I also really dont understand so the information above that you've given me may be of better use on that instead?? However the notation I've got for that one at the start is

"The centre of G is Z(G)= {g e G : ga=ag, for all a e G" which isn't the same as above either
The centre of a group and the centraliser of an element aren't the same thing. (Notice that it's the centre of a group and the centraliser of an element).

The centre of a group is the set of elements which commute with everything in the group. Since the centraliser of an element is the set of elements of the group that commute with that element, this is the same as saying that the centre of a group is the set of elements whose centraliser is the whole group. That is, . Note that this means that a group is abelian if and only if .

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