Quotients of CI-Groups are CI-Groups

We show that a quotient group of a CI-group with respect to (di)graphs is a CI-group with respect to (di)graphs. In [1,2], Babai and Frankl provided strong constraints on which finite groups could be CI-groups with respect to graphs. As a tool in this program, they proved [1, Lemma 3.5] that a quotient group G/N of a CI-group G with respect to graphs is a CI-group with respect to graphs provided that N is characteristic in G. They were not able to prove that a quotient group of a CI-group with respect to graphs is a CI-group with respect to graphs in the general case, and so introduced the notion of a weak CI-group with respect to graphs in order to treat quotient groups of CI-groups. In some sense, the program that Babai and Frankl started was completed by Li [4] when he showed that all CI-groups are solvable. (Babai and Frankl mention in [2] a sequel to their first paper that addressed showing all CI-groups with respect to graphs are solvable. This sequel never appeared.) We will show that a quotient group of a CI-group with respect to (di)graphs is a CI-group with respect to (di)graphs. This will allow for a simplification of the proofs of Babai and Frankl in [1,2] (for example the notion of a weak CI-group with respect to graphs will no longer be needed), and consequently, as Li’s proof in [4] was based on the earlier work of Babai and Frankl, a simplification of the proof that a CI-group with respect to graphs is solvable.