A question about class equation

Question

If $G$ is a group, then class equation is given by

$$|G|=|Z(G)|+ \sum_{i=1}^K[G:C_{G}(x_i)] $$

where $x_i\notin Z(G)$

For dihedral group $D_8$, class equation is given by

$$2+2+2+2$$

but there is another way to write this is

$$1+1+2+2+2$$

My question is: why here two representaions for center of a group is? If order of center is 3( or >3), then is it possible to write as $1+2 $ ? please someone help.Thanks

While it is of course also true that $8=1+3+4$ or $8=\frac12(5^2-3^2)$, those truths wouldn't really represent the class equation — Hagen von Eitzen, Oct 08 '15 at 18:54

score 2 · Answer 1 · edited Nov 05 '19 at 04:21

2

The class equation $$|G|=|Z(G)|+ \sum_{i=1}^K[G:C_{G}(x_i)] $$, $x_i$ is not in the center. Now $|Z(G)|$ is in fact the sum of order of singleton equivalence classes if we go back to the definitions of the defining equivalence relation.

edited Nov 05 '19 at 04:21

Tamas Kanti Garai

304

answered Oct 08 '15 at 18:57

Adelafif

1,275

Adelafif:yes,you are write.you mean there are only two possibility to write center . – user278346 Oct 08 '15 at 19:02

A question about class equation

1 Answers1