Explicit Formulas for the Anti-Trace of 3-by-3 Matrix Powers via Generating Functions

Abstract

Let \textsl{G} be a group and \textsl{H} be a non-normal cyclic subgroup of \textsl{G}. The non-normal cyclic subgroup graph, denoted as $\Gamma^{NN}_{H}(G)$, is defined as a directed graph with vertex set elements of \textsl{G} such that for two distinct elements \textsl{x} and \textsl{y} in \textsl{G}, \textsl{x} is the initial vertex and \textsl{y} is the terminal vertex of an edge if their product is in \textsl{H}. In 2020, the idea of the non-normal subgroup graph, which is the extension of the subgroup graph was developed. This research identifies the non-normal cyclic subgroup graphs connected to order 16 quasidihedral groups. Firstly, the Groups, Algorithms and Programming (GAP) software is used to identify the non-normal cyclic subgroups. The definition of $\Gamma^{NN}_{H}(G)$ is then used to calculate the adjacency and direction of the vertices. Lastly, Maple 2016 software will be used to visualize these graphs.

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