What question did this study set out to answer?

To investigate the existence of cyclic and dihedral NNND and NNN groups and their properties.

April 28, 2026

Cyclic and dihedral NNN-groups and NNND-groups

Key Points

To investigate the existence of cyclic and dihedral NNND and NNN groups and their properties.
Defined NNN Cayley digraphs and explored their properties within cyclic and dihedral groups.
Analyzed conditions under which dihedral groups qualify as NNND or NNN groups.
No cyclic NNND or NNN groups exist.
A dihedral group of order 2n is an NNND or NNN group if n is even, n ≥ 6, and n ≠ 8.

Abstract

Abstract A Cayley digraph on a group 𝐺 is called NNN if the Cayley digraph is normal and its automorphism group contains a non-normal regular subgroup isomorphic to 𝐺. A group is called an NNND-group or an NNN-group if there is an NNN Cayley digraph or graph on the group, respectively. In this paper, it is shown that there is no cyclic NNND-group, and hence no cyclic NNN-group. Furthermore, a dihedral group of order 2 ⁢ n 2n is an NNND-group or an NNN-group if and only if n ≥ 6 n 6 is even and n ≠ 8 n 8.

Cyclic and dihedral NNN-groups and NNND-groups

Key Points

Abstract

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