1
$\begingroup$

Let $C_1,C_2,\ldots,C_n$ be the conjugacy classes of a finite group $G$. It is possible that there are two non-conjugate subgroups $K \leq G$ and $H \leq G$ such that $|H \cap C_i|=|K \cap C_i|$ for all $i = 1,\ldots,n$?

$\endgroup$
6
$\begingroup$

Yes, this is precisely the definition of a Gassmann triple. Your condition is equivalent to the permutation representations $\mathbb{C}[G/K]$ and $\mathbb{C}[G/H]$ being isomorphic. I believe that the smallest example (in terms of $[G:K]$) is $G=SL_3(\mathbb{F}_2)$, $K$ the stabilizer of a line in $\mathbb{F}_2^3$ and $H$ the stabilizer of a plane.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.