# Size of conjugacy class of a quotient group

I am trying to recall what happened to the conjugacy class and centraliser when we quotient out by a normal subgroup.

In particular, we know from Orbit-Stabiliser that

$$|G|=|ccl_G(g)||C_G(g)|$$ for every $$g\in G$$

If we quotient out by a normal subgroup we have that

$$|G|/|N|=|G/N|=|ccl_{G/N}(gN)||C_{G/N}(gN)|$$

My question is what was the relationship between $$|ccl_G(g)|$$ and $$|ccl_{G/N}(gN)|$$ (or $$|C_G(g)|$$ and $$|C_{G/N}(gN)|$$)

Do we need $$N$$ to be a particular subgroup (like a central subgroup, commutator subgroup, etc) to have a special relationship?

Some thoughts:

Certainly, if $$g$$ and $$hgh^{-1}$$ are conjugate elements in $$G$$, then $$(hN)(gN)(hN)^{-1}=hgh^{-1}N$$ so $$gN$$ and $$hgh^{-1}N$$. The question is if $$hgh^{-1}\neq g$$ then $$hgh^{-1}N=gN$$ so $$hgh^{-1}g^{-1}\in N$$ (So, if $$N$$ is a derived subgroup then, the conjugacy class all become $$1$$, i.e. the quotient is abelian. However, what happens in the more general context like if $$N$$ is a central subgroup like $$Z(G)$$?)

Second observation if $$ccl_G(g)$$ is a conjugacy class of size $$1$$, we can't go any smaller so we know that $$|C_{G/N}(gN)|=|C_G(g)|/|N|$$

• What is your $ccl_{G}(g)$? – user549397 Apr 23 at 7:02
• @user549397 I think the OP means the conjugacy class of $g$ – leibnewtz Apr 23 at 7:12
• @leibnewtz Really? Then if $C_G(g)$ is the centralizer of $g$ in $G$, why do we have the first equality? – user549397 Apr 23 at 7:19
• Yes, I mean the conjugacy class. Is there something wrong with the first equality? – daruma Apr 23 at 9:58
• There are (finitely generated) infinite groups with only two conjugacy classes (that is, where every two non-trivial elements are conjugate). See Corollary 1.2 of Denis Osin, Small cancellations over relatively hyperbolic groups and embedding theorems, Ann. Math. (2010). – user1729 Apr 23 at 10:45