Structural Analysis of Artin’s Cokernel for the Group (Q2l×C7)

A fractional Abelian group is a mathematical structure in which operations—typically involving fractions or concepts from fractional calculus—are defined within the context of an Abelian group. In this study, we determine the cyclic decomposition of the fractional Abelian group AC(Q2l×C7). Here, group(Q2l×C7) is the direct product group of the quaternion group Q2l of order 4l and the cyclic group C7 of order 7, then the order of The group(Q2l×C7) is 28l. After knowing Ar(Q2l×C7), the result was as follows: $$A C\left(Q_{2 l}\times C_7\right)=\stackrel{4}{\bigoplus}_{i=1}C_2.$$ The results offer deeper insight into the structural characteristics of the group (Q2l×C7)​, with meaningful implications for both representation theory and computational algebra. Through the analysis of matrix transformations and invariant factor decomposition, this study contributes to a broader understanding of group theory and its applications in mathematical and theoretical contexts. Moreover, the findings may provide a foundational basis for future research in areas such as character theory and modular representations within abstract algebra.