Class equation of normal subgroup

Question

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.

Answer 1

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 $1$.