Erlangen program

Builds on Group Group action Homogeneous space

Definition

Let $X$ be a set and let $G$ be a group with a transitive action on $X$ . In the sense of the Erlangen program, the pair $(X, G)$ is a geometry. A property of subsets of $X$ belongs to the geometry $(X, G)$ if it is invariant under $G$ . This means that, for each subset $S$ and each $g \in G$ , it holds for $S$ exactly when it holds for $g(S)$ . Two subsets of $X$ are equivalent in the geometry if some $g \in G$ maps one onto the other.

Remarks

Example: $X = \R^2$ with $G$ the affine group gives affine geometry. The property "the points $a, b, c$ are collinear" belongs to it, since affine maps send lines onto lines. Non-example: the property "the segment $ab$ has length $1$ " does not belong to affine geometry, since the map $x \mapsto 2x$ is affine and doubles lengths. Since $G$ acts transitively, $X$ is a homogeneous space $G/H$ , where $H$ is the stabilizer of a point.

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Group concept
Group action concept
Homogeneous space concept
Erlangen program definition

Tag	`MSYQ5YEPN5`
URI	`m2://MSYQ5YEPN5`
Kind	definition
Writing	W0 stub
MSC	51N25, 51-03, 20H15
Defines	Erlangen program
Uses	Group, Group action, Homogeneous space

Erlangen program

Definition

Remarks

Reading path

Further reading

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