# Find the class equation for the following groups

Find the class equation for the following groups:

(a) The Quaternion group

(b) $$D_5$$

(c) $$D_6$$

(d) The Klein four group

(a) $$1+1+2+2+2$$

(b) $$1+2+2+5$$

(c) $$1+1+3+3+2+2$$

(d) $$1+1+1+1$$

• They are correct. Not sure this is a useful question, though. Aug 21, 2015 at 5:31
• if you mean by the class equation the size of each conjugacy class, yes. Usually though we put together all of the things in the center, which means all the ones that are 1 are put together, so a would be 2+2+2+2, D would be 4, and C would 2+3+3+2+2
– Alan
Aug 21, 2015 at 6:02
• Can you show how you got these answers, which are by the way correct? Aug 21, 2015 at 6:25

Class equations: $|G|=|Z(G)|+\sum[G:c(a)]$, where $a \notin Z(G)$ and $|cl(a)|=[G:c(a)]$

$Q_8=\{1,-1,i,-i,j,-j,k,-k\}$

$Z(Q_8)=\{1,-1\}$. $c(i)=\{1,-1,i,-i\}$,$|c(i)|=|c(j)|=|c(k)|=4$. Therefore $|cl(i)|=|cl(j)|=|cl(k)|=2$

$|Q_8|=2+2+2+2$.

$D_5=\{e,r,r^2,r^3,r^4,s,rs,r^2s,r^3s,r^4s\}$, $r^5=s^2=e,rs=sr^4$ . $Z(D_5)=\{e\},cl(r)=\{r,r^4\},cl(r^2)=\{r^3,r^2\},cl(s)=\{rs,r^2s,r^3s,r^4s,s\}$

$|D_5|=1+2+2+5$.

$D_6=\{e,r,r^2,r^3,r^4,r^5,s,rs,r^2s,r^3s,r^4s,r^5s\}$, $r^6=s^2=e,rs=sr^5$

$Z(D_6)=\{e,r^3\},cl(r)=\{r,r^5\},cl(r^2)=\{r^2,r^4\},cl(s)=\{s,r^2s,r^4s\},cl(rs)=\{rs,r^3s,r^5s\}$

$|D_6|=2+2+2+3+3$

$V_4=\{e,a,b,c\},a^2=b^2=c^2=1$ (since $V_4$ is abelian group)

$|V_4|=4(Z(V_4))$