For part (b), what does center mean? a commutes with every other element of G.
If a commutes what is gag-1?
So if b ~ a then b = gag-1 = ...
a) Let G be a group. Deﬁne ∼ by the following: a ∼ b ⇐⇒ ∃ g ∈ G such that gag-1 = b.
Prove that ∼ is an equivalence relation.
(b) Suppose a ∈ Z(G). What elements are in the same cell as a with respect to the relation
∽?
(c) Let a ∈ G and deﬁne the centralizer of a, CG(a), as the subset
CG(a) = {g ∈ G : ga = ag}.
Prove that CG(a) ≤ G. If a ∈ Z(G) what can you conclude about CG(a)?
(d) (Bonus) Show that there is a bijective correspondence between elements equivalent (via
∼) to a ∈ G and left cosets of CG(a).
I understand a), I proved ~ is an equivalence relation. But I'm lost on the other three parts. This is not homework by the way per say, it was in my book and I'm studying for our final, this is one of the ones labeled "Very hard" basically. Thank you for your help
Just for clarity, Z(G) indicates the center of the group
For part (c) you need to prove that CG(a) is a subgroup.
That is show
closure: g, h are in CG(a), then is gh in CG(a)?
ga = ag
ha = ah
show gha = agh
inverse: g is in CG(a) is g-1 in CG(a)?
ga = ag
show g-1a=ag-1
(hint g-1a = g-1agg-1)
Also, when a is in Z(G) (a commutes with every element of G), what is CG(a) (all elements of G that commute with a)?
b is only notation for an arbitrary element.
We are saying, consider an arbitrary element b in the cell of a (could be anything we want), and we proved b = a (always no matter how we picked b). This means that any element in the cell of a is equal to a.
Hence, a is the only element in the cell. gag-1 cannot be anything but a.
Alright, so I guess suppose that (G,m) is a group with underlying set G and multiplication operation m:G×G→G. A group (H,m') is said to be a subgroup of (G,m), if H is a subset of G and m' is the restriction of m to H×H.
If we know that (G,m) is a group, and that H is a subset of G, we can define m' to be the restriction of m to H×H. To prove that (H,m') is a subgroup, we need to show that
1. m':H×H→H
2. m' is asociative
3. e is in H.
4. For every x in H, x1 is in H.
But how exactly do I get it down from here
You already have everything. Just disect the definitions.
H is CG(a) in this case. The operation m' is just * (written g*h or just gh).
I will disect 1 in detail for you:
We want to show * : CG(a) x CG(a) -> CG(a)
This means that for any g, h in CG(a)
We need to show gh or g * h is in CG(a)
g in CG(a) means what? ga = ag (commutative)
similarly ha = ah
How do we show gh is in CG(a)?
We need to show (gh)a = a(gh)
How? (gh)a ={associative}= g(ha) ={h in CG(a)}= g(ah) ={assoc}= (ga)h ={g in CG(a)}= (ag)h ={assoc}= a(gh)
We can assume associativity (see below)
This proves 1
Now 2, associativity, is trivial, because any g,h,k in CG(a) is also in G which is already associative by being a group.
For 3, what is e? How do we show e is in CG(a)?
For 4, what is x^{-1}? If x is in CG(a), how can we show x^{-1} is in CG(a)?
Already provided the hint in above post (this one is a bit tricky).
After 4 you are done
Alrighty, so e is the identity element of the group G. It's in CG(a) because ea = ae (=e) right?
As for the inverse, how does this compare to what you did and is it legal...
We can assume that x-1a = x-1a
x-1a = x-1a
x-1a = x-1ae
x-1a = x-1a(xx-1)
x-1a = ax-1(xx-1)
x-1a = a(x-1x)x-1
x-1a = aex-1
x-1a = ax-1