If in a group there are only two conjugacy classes then group is isomorphic to Z2 . This statement is true or not ?
Asked
Active
Viewed 22 times
1
-
Hint for finite groups: cardinality of conjugacy class divides order of group. – drhab Jun 10 '17 at 10:12
-
The simple answer is no, it is not true. – Derek Holt Jun 10 '17 at 10:56
1 Answers
0
Yes the statement is true for finite groups!!
First observe that for $\mathbb{Z}_{2}$ the statement is correct. Now for any group $G$ let us assume it has only 2 conjugacy classes. Now the identity element $\{\ e \}\ $ has its own singleton class. Hence the other remaining class contains $|G|-1$ number of elements. But the size of conjugacy class divides the order of the group, hence $|G|-1 | |G|$. What is the only possibility for $|G|$ in this case ; |G|=2 is the only possible one. Hence the result.
Riju
- 4,015
-
-
Yeah thanks!!! I assumed he was only asking about finite groups.. Although I didn't knew about infinite groups!! – Riju Jun 10 '17 at 10:27