# Semiregular subgroups of automorphism group of cayley graphs

Let $\Gamma$ be a Cayley graph over group $K$ and $H$ be a semiregular subgroup of $Aut(\Gamma)$ with two orbits. Then $|K|=2|H|$. Is there any other relation between $H$ and $K$ in general? What about special cases?

• In order to conclude that $|K|=2|H|" you should add WITH TWO ORBITS at the end of your first sentence. Now that your question has changed my earlier answer is not applicable and I have deleted it. – Nick Gill Nov 11 '12 at 16:17 ## 1 Answer In general, you can't say mauch. Take$\Gamma$to be the complete graph$K_n$. Then$\Gamma$is a Cayley graph for any group$K$of order$n$, and any group$H$of order$n/2$acts semiregularly on$\Gamma\$ with two orbits.

• Ok. Thanks so much. Do you think may be any relation between this two subgroups in special cases? For example for (non-complete)Cayley graphs of order twice a prime? – majid arezoomand Nov 20 '12 at 13:10