An Introduction to Commutative Trinary Groups

This paper is an introduction to the notion of a trinary group: a set with a trinary operation. The paper was hastily written for my UROP over the summer of 2010 to put together some of their properties so that I could perhaps use the ideas for my research (see Classification of quadratic iteration graphs).


In this document, we will define the notion of a commutative trinary group, which is group-like algebraic object with an associated trinary operation, show basic properties of these objects, and determine the trinary subgroups of Fp* useful for understanding the basic structure of quadratic residue graphs. We will only talk about trinary groups which are commutative 1) because of the difficulties inherent in non-commutative algebra and 2) because the motivation in studying trinary groups is to understand quadratic residues in Fp*.

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