Binary operation

This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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Definition

A binary operation on a set $X$ is a function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ast: X \times X \to X} .

Examples

• Addition is a binary operation on the integers, rational numbers, real numbers and complex numbers.
• Matrix multiplication is a binary operation on the set of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \times n} matrices for any Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} .

In abstract algebra

A group is a set equipped with a binary operation that has an identity element, an inverse for each element, and associativity.