Suppose $p$ and $q$ are two prime divisor of the order of a finite group $G$. I want to know if $G$ has an element of order $pq$ using the character table of $G$. Is this possible? If so, please suggest a method.
Asked
Active
Viewed 61 times
2
-
Such an element always exists in abelian groups, but not necessarily in non-abelian groups (e.g. $S_3$). I know of a proof, but unfortunately it doesn't use the character table of the group. – Kaj Hansen Jul 07 '14 at 09:46
-
Not sure how much of this can be seen from the character table, but the existence of said element is equivalent to the existence of elements of orders $p$ and $q$ which commute (which is then to some extend related to how the conjugacy classes look, so being able to tell from the character table is not completely improbable). – Tobias Kildetoft Jul 07 '14 at 10:09
-
So if there is a method to find out existence of said elements in any finite group even without usage of character table what will be that method? – Sahar rad Aug 11 '14 at 06:26