Generalized quaternion NCI-groups, NNN-groups and NNND-groups
Jun-Feng Yang, jia-Li Du, Yan-Quan Feng, Young Soo Kwon

TL;DR
This paper classifies generalized quaternion groups as NCI-groups for all n≥2 and determines their properties related to NNN and NNND groups, expanding understanding of Cayley graph symmetries.
Contribution
It proves that all generalized quaternion groups are NCI-groups and characterizes when they are NNND-groups, solving open classification questions.
Findings
Q_{4n} is an NCI-group for all n≥2.
Q_{4n} is not an NNN-group for any n≥2.
Q_{4n} is an NNND-group if and only if n≥6 and n is even.
Abstract
A Cayley (di)graph of a finite group is called CI if, for every Cayley (di)graph of , implies that for some . The group is called an NDCI-group (resp. NCI-group) if every normal Cayley digraph (resp. graph) of is CI. It was shown that the generalized quaternion group of order () is an NDCI-group if and only if either or is odd, but its NCI-group classification has been left as an open question. In this paper, we solve the question and prove that is an NCI-group for every . A normal Cayley (di)graph of a group is called NNN if its automorphism group contains a non-normal regular subgroup isomorphic to , and is called an NNND-group (resp. NNN-group) if it admits an NNN Cayley digraph (resp. graph). In this paper, we show that…
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