# Embedding finite groups in symmetric groups whilst preserving conjugacy

Let $G$ be a finite group and suppose that $f,g \in G$ are not conjugate in $G$. It is classical result that every finite group embeds into a finite symmetric group $S = S_{|G|}$. My question is: can this embedding be modified in a way so that $f \not\sim_G g$?

It can be easily seen that in general, the answer is negative - consider $C_p$, a cyclic group of prime order. However, is there some nice description for instances, when it will be possible?

• Do you mean $f \not\sim_S g$? Also "can this embedding be modified" is not clear. What kinds of modifications do you allow? – Derek Holt Feb 15 '16 at 17:49
• Basically, anything is fine as long as I remain within the class of finite symmetric groups. Of course I still require the map to be a monomorphism. – Michal Ferov Feb 15 '16 at 21:54

The class of groups (which can be embedded into an $S_n$ such that in the embedded image conjugacy is completely determined by cycle structure --lets call them CCS-groups for conjugacy cycle structure'') will be closed under sub direct products (duplicate, triplicate etc. the cycles in additional copies), so it is more than just $S_n$. CCS-groups also include (explicit test) the dihedral groups of order 8 and 12. (From trying out groups of small order, and using the following condition, I conjecture that this is all.)
The same argument as for $C_p$ (coprime powers must be conjugate) shows that a CCS-group must have an integral character table. (See https://mathoverflow.net/questions/134581/groups-in-which-all-characters-are-rational for this separate question) This property is necessary but not sufficient ($Q_8$ has an integral character table but is not CCS).