Euclid's view and Klein's view of Geometry and Associativity in Group"Euclid's view and Klein's view of Geometry and Associativity in Group One common item in the have a look at of Euclidean geometry (Euclid's view) is "congruence" relation- specifically ""congruence of triangles"". We recognize that this congruence relation is an equivalence relation: Every triangle is congruent to itself If triangle T_1 is congruent to triangle T_2 then T_2 is congruent to T_1. If T-1 is congruent to T_2 and T_2 is congruent to T_3, then T_1 is congruent to T_3.

Janet Hart

Janet Hart

Answered question

2022-09-19

Euclid's view and Klein's view of Geometry and Associativity in Group
One common item in the have a look at of Euclidean geometry (Euclid's view) is "congruence" relation- specifically ""congruence of triangles"". We recognize that this congruence relation is an equivalence relation
Every triangle is congruent to itself
If triangle T 1 is congruent to triangle T 2 then T 2 is congruent to T 1 .
If T 1 is congruent to T 2 and T 2 is congruent to T 3 , then T 1 is congruent to T 3 .
This congruence relation (from Euclid's view) can be translated right into a relation coming from "organizations". allow I s o ( R 2 ) denote the set of all isometries of Euclidean plan (=distance maintaining maps from plane to itself). Then the above family members may be understood from Klein's view as:
∃ an identity element in I s o ( R 2 ) which takes every triangle to itself.
If g I s o ( R 2 ) is an element taking triangle T 1 to T 2 , then g 1 I s o ( R 2 ) which takes T 2 to T 1 .
If g I s o ( R 2 ) takes T 1 to T 2 and g I s o ( R 2 ) takes T 2 to T 3 then h g I s o ( R 2 ) which takes T 1 to T 3 .
One can see that in Klein's view, three axioms in the definition of group appear. But in the definition of "Group" there is "associativity", which is not needed in above formulation of Euclids view to Kleins view of grometry.
Question: What is the reason of introducing associativity in the definition of group? If we look geometry from Klein's view, does "associativity" of group puts restriction on geometry?

Answer & Explanation

Klecanlh

Klecanlh

Beginner2022-09-20Added 11 answers

Associativity is implicitly present in Klein's formulation since composition of maps is associative: if f maps w to x, g maps x to y, and h maps y to z, then gf maps w to y and hg maps y to z. So (hg)f has the same result as h(gf): w is sent to x by f, which is then sent to z by hg; w is sent to y by gf, which is then sent to z by h.
The study of groups grew out of the study of sets of permutations closed under composition. So again, the study of groups started out as the study of composition of certain maps. The group axioms were designed to be an abstract description of composition of maps, and so it was natural to require associativity.

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