Changes

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.

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 <math>G.\!</math>

Tables&nbsp;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 <math>x_i\!</math> in <math>G\!</math> is a function from <math>G\!</math> to <math>G\!</math> that is formed by considering the effects of <math>x_i\!</math> on the elements of <math>G\!</math> when <math>x_i\!</math> acts in the role of the first operand of the group operation.  Notating this function as <math>h_1(x_i) : G \to G,\!</math> the '''regular ante-representation''' of <math>G\!</math> is a map <math>h_1 : G \to (G \to G)\!</math> that is schematized in Table&nbsp;32.2.  Here, each of the functions <math>h_1(x_i) : G \to G\!</math> is represented as a set of ordered pairs of the form <math>(x_j ~,~ x_i * x_j).\!</math>

The '''ante-representation''' of <math>x_i\!</math> in <math>G\!</math> is a function from <math>G\!</math> to <math>G\!</math> that is formed by considering the effects of <math>x_i\!</math> on the elements of <math>G\!</math> when <math>x_i\!</math> acts in the role of the first operand of the group operation.  Notating this function as <math>h_1(x_i) : G \to G,\!</math> the '''regular ante-representation''' of <math>G\!</math> is a map <math>h_1 : G \to (G \to G)\!</math> that is schematized in Table&nbsp;32.2.  Here, each of the functions <math>h_1(x_i) : G \to G\!</math> is represented as a set of ordered pairs of the form <math>(x_j ~,~ x_i * x_j).\!</math>

In order to have concrete materials available for future discussions of group theoretic issues, the remainder of this section takes up a pair of small examples, the groups of order <math>4,\!</math> and uses them to illustrate the chain of definitions and the forms of representation given above.

In order to have concrete materials available for future discussions of group theoretic issues, the remainder of this section takes up a pair of small examples, the groups of order <math>4,\!</math> and uses them to illustrate the chain of definitions and the forms of representation given above.

There are just two groups of order <math>4.\!</math>  Both are abelian (commutative), but one is cyclic and the other is not.  The cyclic group on <math>4\!</math> elements is commonly referred to as <math>Z_4.\!</math>  (The German words ''Zahl'' for &ldquo;number&rdquo; and ''Zyklus'' for &ldquo;cycle&rdquo; together make the notation <math>Z_n\!</math> suggestive of the integers modulo <math>n,\!</math> which form a cyclic group of order <math>n\!</math> under the addition operation.)  The acyclic group on <math>4\!</math> elements is usually called the ''Klein 4 group'' and notated as <math>V_4.\!</math>  (The German word ''Vierbein'' is the substantive form of an adjective that means &rdquo;four-legged&rdquo;.)

There are just two groups of order <math>4.\!</math>  Both are abelian (commutative), but one is cyclic and the other is not.  The cyclic group on <math>4\!</math> elements is commonly referred to as <math>Z_4.\!</math>  (The German words ''Zahl'' for &ldquo;number&rdquo; and ''Zyklus'' for &ldquo;cycle&rdquo; together make the notation <math>Z_n\!</math> suggestive of the integers modulo <math>n,\!</math> which form a cyclic group of order <math>n\!</math> under the addition operation.)  The acyclic group on <math>4\!</math> elements is usually called the ''Klein 4 group'' and notated as <math>V_4.\!</math>  (The German word ''Vierbein'' is the substantive form of an adjective that means &ldquo;four-legged&rdquo;.)

For the sake of comparison, I give a discussion of both these groups.

For the sake of comparison, I give a discussion of both these groups.  However, because it figures more prominently in another part of the present construction, I discuss V4 first and foremost.

The next series of Tables presents the group operations and regular representations for the groups V4 and Z4.  If a group is abelian, as both of these groups are, then its h1 and h2 representations are indistinguishable, and a single form of regular representation h : G > (G >G) will do for both.

The next series of Tables presents the group operations and regular representations for the groups <math>V_4\!</math> and <math>Z_4.\!</math> If a group is abelian, as both of these groups are, then its <math>h_1\!</math> and <math>h_2\!</math> representations are indistinguishable, and a single form of regular representation <math>h : G \to (G \to G)\!</math> will do for both.

Tables 33.1 shows the multiplication table of the group V4, while Tables 33.2 and 33.3 present two versions of its regular representation.  The first version, somewhat hastily, gives the functional representation of each group element as a set of ordered pairs of group elements.  The second version, more circumspectly, gives the functional representative of each group element as a set of ordered pairs of element names, also referred to as "objects", "points", "letters", or "symbols".

Tables 33.1 shows the multiplication table of the group V4, while Tables 33.2 and 33.3 present two versions of its regular representation.  The first version, somewhat hastily, gives the functional representation of each group element as a set of ordered pairs of group elements.  The second version, more circumspectly, gives the functional representative of each group element as a set of ordered pairs of element names, also referred to as "objects", "points", "letters", or "symbols".