# Class equation of normal subgroup

Can someone give me an example of the class equation applied to the normal subgroup of a group?

For instance say I have a binary operation $$\star: G \times N\rightarrow N$$, where $$N={\{n_i\}}$$ and $$N \trianglelefteq G$$, such that for $$g \in G$$, $$g \star n_i$$ = $$g n_i g^{-1} \in N$$. Now how would I go about applying the class equation here, generally given as $$|G|$$ = $$|Z(G)|$$ + $$\sum$$[$$G$$:$$C_G$${$$x_i$$)] ? All my examples seem to deal with the group $$G$$ acting on itself as a whole, and not on a particular subgroup.

• For this case see for example here. You have to replace $Z(G)$ by $S_0$ and $C_G(x_i)$ by the corresponding set. So the $N$ must appear in your formula. – Dietrich Burde Mar 8 '20 at 9:24

A group $$G$$ can act on itself by conjugacy, meaning that it exists a map $$G \times G \to G$$, defined by $$(g,a) \mapsto gag^{-1}$$, which fulfills action properties; in fact:

• $$(e,a)=eae^{-1}=a, \forall a \in G$$;
• $$(g,(h,a))=g(h,a)g^{-1}=g(hah^{-1})g^{-1}=(gh)a(gh)^{-1}=(gh,a), \forall g,h,a \in G$$.

The stabilizer of $$a \in G$$ is:

$$\operatorname{Stab}(a)=\{g \in G \mid gag^{-1}=a\}=C_G(a) \tag 1$$

The Orbit-Stabilizer Theorem states that:

$$|O(a)||\operatorname{Stab}(a)|=|G|, \forall a \in G \tag 2$$

By $$(1)$$, $$(2)$$ reads:

$$|O(a)|=\frac{|G|}{|C_G(a)|}, \forall a \in G \tag 3$$

Since the set of orbits is a partition of the acted set, namely $$G$$ in this case, we have:

$$|G|=\sum_{a \in \{orbits \space rep's\}}|O(a)|=\sum_{a \in \{orbits \space rep's\}}\frac{|G|}{|C_G(a)|} \tag 4$$

So, the orbits of the action of $$G$$ on itself by conjugacy are just the conjugacy classes of $$G$$, and $$(4)$$ is the Class Equation popping up when considering conjugacy as an action of $$G$$ on itself, rather than an equivalence relation on $$G$$.

Now, if $$N \unlhd G$$, then conjugacy establishes an action of $$G$$ on $$N$$; in fact, $$gNg^{-1} \subseteq N, \forall g \in G$$ (by normality), and action properties are verified as above. So, if you go through the steps $$(1)$$ to $$(4)$$ above in this different case, you'll come up with:

$$|N|=\sum_{n \in \{orbits \space rep's\}}|O(n)|=\sum_{n \in \{orbits \space rep's\}}\frac{|G|}{|C_G(n)|} \tag 5$$

where here $$\{orbits\space rep's\} \subseteq N$$. Note that, by $$(3)$$, $$|O(n)|=1 \Leftrightarrow C_G(n)=G \Leftrightarrow n \in Z(G)$$; therefore $$(5)$$ can be furtherly worked out into:

$$|N|=|N \cap Z(G)|+\sum_{n \in \{Orbits \space rep's\}}\frac{|G|}{|C_G(n)|} \tag 6$$

where "$$Orbits$$" (capital "O") are the orbits of size greater than $$1$$.