Euclid's view and Klein's view of Geometry and Associativity in GroupOne 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 relationEvery triangle is congruent to itselfIf 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?

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Euclid&#039;s view and Klein&#039;s view of Geometry and Associativity in GroupOne common item in the have a look at of Euclidean geometry (Euclid&#039;s view) is &quot;congruence&quot; relation- specifically &quot;&quot;congruence of triangles&quot;&quot;. We recognize that this congruence relation is an equivalence relationEvery triangle is congruent to itselfIf 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&#039;s view) can be translated right into a relation coming from &quot;organizations&quot;. 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&#039;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&#039;s view, three axioms in the definition of group appear. But in the definition of &quot;Group&quot; there is &quot;associativity&quot;, 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&#039;s view, does &quot;associativity&quot; of group puts restriction on geometry?

Klecanlh · Accepted Answer

Associativity is implicitly present in Klein&#039;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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