The Center of a Group as the centralizer of a subgroup.

Oct 2009
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I'm sure this is true. I'm not sure this is the only such subgroup, and I'm not sure this is the subgroup they're thinking of. What do you think?

If notation isn't clear:

\(\displaystyle N(g) = \{n \in G : ng = gn\}\)
 
Oct 2009
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I'm sure this is true. I'm not sure this is the only such subgroup, and I'm not sure this is the subgroup they're thinking of. What do you think?

If notation isn't clear:

\(\displaystyle N(g) = \{n \in G : ng = gn\}\)

But what \(\displaystyle g\in G\) do you choose for \(\displaystyle N(g)\) ?!? This should make it clear your choice isn't correct...

I think the answer is most trivial: \(\displaystyle Z(G) =C(G)\) , so the center of a group G is the centralizer of the subgroup G of G...**sigh**...yes, I know: it's lame.

Tonio
 
Oct 2009
195
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But what \(\displaystyle g\in G\) do you choose for \(\displaystyle N(g)\) ?!? This should make it clear your choice isn't correct...

I think the answer is most trivial: \(\displaystyle Z(G) =C(G)\) , so the center of a group G is the centralizer of the subgroup G of G...**sigh**...yes, I know: it's lame.

Tonio
Sorry, I meant for all g in G. That is, for any n that is not in any N(g). I think thats true, since if n is in N(g), ng = gn for some g in G, so the result is trivial.

Your answer is obvious. It's certainly true, but I hardly see why they'd want do highlight this particular point. But it is the most logical choice.
 
Oct 2009
4,261
1,836
Sorry, I meant for all g in G. That is, for any n that is not in any N(g). I think thats true, since if n is in N(g), ng = gn for some g in G, so the result is trivial.


I think you're confused (or perhaps I am: go figure!) : you define \(\displaystyle N(g):=\{n\in G\;;\;ng=gn\}=:C(g)=\) the centralizer of the element \(\displaystyle g\in G\) (try no to use the letter N for this since it is used for another thing, the normalizer). But an element \(\displaystyle z\in Z(G)\) belongs to the centralizer of each and every element of G! So it can't be that an element in the center is NOT in some element's centralizer

Tonio


Your answer is obvious. It's certainly true, but I hardly see why they'd want do highlight this particular point. But it is the most logical choice.
.
 
Oct 2009
195
19
After looking up wikipedia, it seems my book wanted me to learn them backwards. Here is what my book indicated to me:

The normalizer of a in G is a set \(\displaystyle N(a) = \{g \in G : ag = ga\}\)

The centralizer of a subset \(\displaystyle H \subset G\) is a set \(\displaystyle C(H) = \{g \in G : gh = hg, \forall h \in H\}
\)

According to Wikipedia, I've got this backwards.