Changes

If nothing more succinct is available, a group can be specified by means of its ''operation table'', usually styled either as a ''multiplication table'' or an ''addition table''.  Table&nbsp;32.1 illustrates the general scheme of a group operation table.  In this case the group operation, treated as a &ldquo;multiplication&rdquo;, is formally symbolized by a star <math>(*),\!</math> as in <math>x * y = z.\!</math>  In contexts where only algebraic operations are formalized it is common practice to omit the star, but when logical conjunctions (symbolized by a raised dot <math>(\cdot)\!</math> or by concatenation) appear in the same context, then the star is retained for the group operation.

If nothing more succinct is available, a group can be specified by means of its ''operation table'', usually styled either as a ''multiplication table'' or an ''addition table''.  Table&nbsp;32.1 illustrates the general scheme of a group operation table.  In this case the group operation, treated as a &ldquo;multiplication&rdquo;, is formally symbolized by a star <math>(*),\!</math> as in <math>x * y = z.\!</math>  In contexts where only algebraic operations are formalized it is common practice to omit the star, but when logical conjunctions (symbolized by a raised dot <math>(\cdot)\!</math> or by concatenation) appear in the same context, then the star is retained for the group operation.

Another way of approaching the study or presenting the structure of a group is by means of a ''group representation'', in particular, one that represents the group in the special form of a ''transformation group''.  This is a set of transformations acting on a concrete space of &ldquo;points&rdquo; or a designated set of &ldquo;objects&rdquo;.  In providing an abstractly given group with a representation as a transformation group, one is seeking to know the group by its effects, that is, in terms of the action it induces, through the representation, on a concrete domain of objects.  In the type of representation known as a ''regular representation'', one is seeking to know the group by its effects on itself.

Another way of approaching the study or presenting the structure of a group is by means of a "group representation", in particular, one that represents the group in the special form of a "transformation group".  This is a set of transformations acting on a concrete space of "points" or a designated set of "objects".  In providing an abstractly given group with a representation as a transformation group, one is seeking to know the group by its effects, that is, in terms of the action it induces, through the representation, on a concrete domain of objects.  In the type of representation known as a "regular representation", one is seeking to know the group by its effects on itself.

Tables 32.2 and 32.3 illustrate the two conceivable ways of forming a regular representation of a group G.

Tables 32.2 and 32.3 illustrate the two conceivable ways of forming a regular representation of a group <math>G.\!</math>

The "ante representation" of xi in G is a function from G to G that is formed by considering the effects of xi on the elements of G when xi acts in the role of the first operand of the group operation.  Notating this function as "h1(xi) : G >G", the "regular ante representation" of G is a map h1 : G  > (G >G) that is schematized in Table 32.2.  Here, each of the functions h1(xi) : G >G is represented as a set of ordered pairs of the form <xj, xi*xj>.

The "ante-representation" of xi in G is a function from G to G that is formed by considering the effects of xi on the elements of G when xi acts in the role of the first operand of the group operation.  Notating this function as "h1(xi) : G >G", the "regular ante representation" of G is a map h1 : G  > (G >G) that is schematized in Table 32.2.  Here, each of the functions h1(xi) : G >G is represented as a set of ordered pairs of the form <xj, xi*xj>.

The "post representation" of xi in G is a function from G to G that is formed by considering the effects of xi on the elements of G when xi acts in the role of the second operand of the group operation.  Notating this function as "h2(xi) : G >G", the "regular post representation" of G is a map h2 : G  > (G >G) that is schematized in Table 32.3.  Here, each of the functions h2(xi) : G >G is represented as a set of ordered pairs of the form <xj, xj*xi>.

The "post-representation" of xi in G is a function from G to G that is formed by considering the effects of xi on the elements of G when xi acts in the role of the second operand of the group operation.  Notating this function as "h2(xi) : G >G", the "regular post representation" of G is a map h2 : G  > (G >G) that is schematized in Table 32.3.  Here, each of the functions h2(xi) : G >G is represented as a set of ordered pairs of the form <xj, xj*xi>.

   

   

Table 32.1  Scheme of a Group Multiplication Table

Table 32.1  Scheme of a Group Multiplication Table