Groups: a matter of relationships

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.

Written by

Daniel Lignon

PublishedNovember 18, 2021UpdatedMay 16, 20261 min read

Take a line segment. Which plane geometric transformations leave it invariant, meaning unchanged? One is the reflection across the line containing the segment (denoted S_D), which leaves it pointwise invariant; another is the reflection across the perpendicular bisector of the segment (denoted S_M), which interchanges its points while leaving it invariant as a whole.

What happens when we compose them (an operation denoted

\circ

), applying one transformation after the other? The situation can be summarized in the table below, known as the operation's Cayley table.

Now for numbers ----------------------------

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