Combinatorial invariants for certain classes of non-abelian groups

This article focuses on the study of zero-sum invariants of finite non-abelian groups. We address two main problems: the first centers on the ordered Davenport constant and the second on Gao's constant. We establish a connection between the ordered Davenport constant and the small Davenport constant for a finite non-abelian group of even order, which in turn gives a relation with the Noether number. Additionally, we confirm a conjecture of Gao and Li for a non-abelian group of order $2p^α$, where $p$ is a prime. Furthermore, we prove a conjecture that connects the ordered Davenport constant to the Loewy length for certain classes of finite $2$-groups.

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