On a Note From Oliver Concerning Generalized Euclidean Group Rings

Abstract

In an unpublished 1987 letter, Bob Oliver determined which elementary abelian 2-groups have generalized euclidean integral group rings. He produced a filtration of E ₂(R) by normal subgroups, sandwiched between elementary and special linear relative groups, with layers that are second homology groups with mod-2 coefficients. His proof is presented here, with related consequences for some other finite groups.

Key Words:

• Congruence subgroup
• Elementary matrices
• Euclidean
• Group ring
• K ₁-stability
• Special linear group

2010 Mathematics Subject Classification:

• Primary 19B14
• Secondary 13F07, 16S34, 19J10, 20H05

ACKNOWLEDGMENT

I wish to thank James McNamara, who checked the details of the proof of Theorem 7, and presented it in a talk as part of his Master's Degree work at Miami University.

Notes

Communicated by S. Sehgal.