The concept of a group first emerged from efforts to solve equations in the 19th century and soon became indispensable, highlighting parallels between situations that at first seem quite different. Let's see why mathematicians are so group-minded.
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The concept of a group emerged in the early 19th century to solve polynomial equations and very quickly spread to other fields, including those outside mathematics: it allows us to focus on relationships between objects rather than on the objects themselves.
The first examples that spring to mind are groups of numbers. But groups crop up in every area.
When we first start working with groups, we patiently draw up the tables for those with only a few elements. A one-element group consists solely of the identity element and is therefore unique. Similarly, groups with two or three elements are unambiguously determined. The surprises begin with four elements...
In geometry, what could be more natural than to seek, for each object, the transformations that leave it unchanged? This means dissecting those transformations and identifying the object's symmetries—and, surprise, groups emerge quite naturally...
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