Bass's conjecture on Davenport and Gao constants of metacyclic groups

Let CnC_n be the cyclic group of order nn, and let

Gm,n,s=CnsCm=x,yxm=yn=1, yx=xysG_{m,n,s}=C_n\rtimes_s C_m=\langle x,y\mid x^m=y^n=1,\ yx=xy^s\rangle

be a metacyclic group. For triples (m,n,s)N×N×Z(m,n,s)\in\mathbb N\times\mathbb N\times\mathbb Z with multiplicative order ordn(s)=m\operatorname{ord}_n(s)=m, write d(G){\sf d}(G) for the small Davenport constant and E(G){\sf E}(G) for the Gao constant. Bass's conjecture. For every such triple,

d(Gm,n,s)=m+n2andE(Gm,n,s)=mn+m+n2.{\sf d}(G_{m,n,s})=m+n-2\quad\text{and}\quad {\sf E}(G_{m,n,s})=mn+m+n-2.

These equalities attain the general lower bounds for the metacyclic group Gm,n,sG_{m,n,s}; they are known for dihedral and dicyclic groups and for groups Gp,q,sG_{p,q,s} with pp and qq prime, but the conjecture is otherwise presented here as unresolved.

Sources & referencesView supporting material

Primary source

Danilo Vilela Avelar, Fabio Enrique Brochero Martínez and Sávio Ribas, “A note on Bass' conjecture”, arXiv:2302.11754 (2023).

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