A note on one decomposition of the 2-primary part of K₂(ℤ[C_p × (C₂)ⁿ])

Abstract

Let C_n be a cyclic group of order n. We investigate K₂ of integral group rings via the Mayer-Vietoris sequence, and give a decomposition of the 2-primary torsion subgroup of $K_2(\mathbb{Z}[C_p\times {(C_2)}^n])$ for any prime $p\equiv 3,5,7\hspace{1em}(\mathrm{mod}8)$, in particular, $K_2(\mathbb{Z}[C_3\times {(C_2)}^n])$ is proven to be a finite abelian 2-group. As an application, we prove $K_2(\mathbb{Z}[C_3\times {(C_2)}^2])$ is an elementary abelian 2-group of rank at least 14, at most 16.

KEYWORDS:

• Dennis-Stein symbol
• Group rings
• K₂-group
• Mayer-Vietoris sequence

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 16S34
• 19C40
• 19C99

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.

Acknowledgments

We thank the referees for their time and comments.

Notes

1 There exists a calculation error in the last paragraph on p. 4 in [20]. As for the specific reason for the error, please refer to Scholium in Section 2.2 and Remark 4.5 of this paper for more details. Hence, $K_2(\mathbb{Z}[C_2\times C_2])$ is an elementary abelian 2-group of rank at least 6, at most 7.