---
tag: 2DT5DGTB8V
kind: definition
name: Bachmann's reflection geometry
msc: [51F15, 51F05]
defines: J0K6T02FQQ
uses: [G5S1M9, 06BRDF3F0E, VKFPCSRNDH]
sources:
  - url: https://en.wikipedia.org/wiki/Absolute_geometry
---

## Definition

Let $G$ be a [[group|G5S1M9]] and let $S$ be a set of involutions of $G$, that is, of elements of order $2$.
Assume that $S$ generates $G$ and is closed under conjugation by elements of $G$.
Call the elements of $S$ *lines*, written $a, b, c, g, h, j$.
For $\sigma, \tau \in G$, write $\sigma \mid \tau$ if $\sigma \tau$ is an involution.
Call the elements $ab$ with $a, b \in S$ and $a \mid b$ *points*, written $P, Q$.
For a point $P$ and a line $g$, $P \mid g$ means that $P$ lies on $g$.
For lines $a$ and $b$, $a \mid b$ means that they are perpendicular.
The pair $(G, S)$ is a *metric plane* if:

1. for any points $P, Q$ there is a line $g$ with $P \mid g$ and $Q \mid g$;
2. if $P \mid g$, $P \mid h$, $Q \mid g$ and $Q \mid h$, then $P = Q$ or $g = h$;
3. if $a, b, c \mid P$ for a point $P$, then $abc \in S$;
4. if $a, b, c \mid g$ for a line $g$, then $abc \in S$;
5. there are lines $g, h, j$ with $g \mid h$ such that none of $j \mid g$, $j \mid h$, $j \mid gh$ holds.

## Remarks

Clauses 3 and 4 are the three reflections theorems.
The product of the reflections in three lines through a common point, or in three lines perpendicular to a common line, is again a reflection.
Clause 5 excludes degenerate cases.

Example: let $G$ be the group of isometries of the Euclidean plane generated by the [[reflections|06BRDF3F0E]] in lines, and $S$ the set of these reflections.
Two reflections compose to an involution exactly when their lines are perpendicular, and then the product is the half-turn about the intersection point.
A half-turn about $P$ times the reflection in $g$ is an involution exactly when $P$ lies on $g$.
The product of reflections in three concurrent lines, or in three parallel lines, is a reflection, which gives clauses 3 and 4.

Non-example: let $G$ be cyclic of order $2$ and $S$ its non-identity element $s$.
Then $s s = 1$ is not an involution, so there are no points and clause 5 fails.

Metric planes give a group-theoretic form of plane [[absolute geometry|VKFPCSRNDH]].
The real hyperbolic plane and the real elliptic plane give further examples, with the same construction from their reflections.