Unexpected groups

Abstract

The reader should first try the following exercise, which motivates the later work. Exercise: Prove that the set of real matrices forms a group under the usual matrix multiplication. This exercise comes with a guarantee that it will provoke at least a doubletake when first encountered. The usual reaction is to protest that these matrices are obviously singular, or to complain that the identity is missing from the set. Of course those familiar with this example can feel suitably smug in their rapid dismissal of these criticisms, but what was your reaction the first time you saw it?

103.33 Conditions for the terms of an arithmetic sequence to form a multiplicative modular group

“…For each integer , the set of all positive integers less than and relatively prime to is a group under multiplication modulo . Many authors, for example, Brakes [1], Denniss [2], McLean [3], and Hidayah and Purwanto [4], have studied these groups. Denniss constructs groups in modular arithmetic under multiplication modulo each of which has an identity element which is not necessarily 1, for some value of .…”

Singular matrices applied to 3 × 3 magic squares

“…The topic of singular matrices is not encountered often in the pedagogical literature or in books on the recreational aspects of mathematics. Other articles of interest here may be [2,3]. My interest in magic squares was recently rekindled by D. B. Eperson's short note [4].…”