Changes

In following these maps, notice how closely one is treading in these representations to defining each element in terms of itself, but without quite going that far.  There are a couple of catches that save this form of representation from falling into a &ldquo;vicious circle&rdquo;, that is, into a pattern of self-reference that would beg the question of a definition and vitiate its usefulness as an explanation of each group element's action.  First, the regular representations do not represent that a group element is literally ''equal to'' a set of ordered pairs involving that very same group element, but only that it is ''mapped to'' something like this set.  Second, careful usage would dictate that the ''something like'' that one finds in the image of a representation, being something that is specified only up to its isomorphism class, is a transformation that really acts, not on the group elements <math>x_j\!</math> themselves, but only on their inert tokens, inactive images, partial symbols, passing names, or transitory signs of the form <math>{}^{\backprime\backprime} x_j {}^{\prime\prime}.\!</math>

In following these maps, notice how closely one is treading in these representations to defining each element in terms of itself, but without quite going that far.  There are a couple of catches that save this form of representation from falling into a &ldquo;vicious circle&rdquo;, that is, into a pattern of self-reference that would beg the question of a definition and vitiate its usefulness as an explanation of each group element's action.  First, the regular representations do not represent that a group element is literally ''equal to'' a set of ordered pairs involving that very same group element, but only that it is ''mapped to'' something like this set.  Second, careful usage would dictate that the ''something like'' that one finds in the image of a representation, being something that is specified only up to its isomorphism class, is a transformation that really acts, not on the group elements <math>x_j\!</math> themselves, but only on their inert tokens, inactive images, partial symbols, passing names, or transitory signs of the form <math>{}^{\backprime\backprime} x_j {}^{\prime\prime}.\!</math>

These reservations are crucial to understanding the form of explanation that a regular representation provides, that is, what it explains and what it does not.  If one is seeking an ontological explanation of what a group and its elements ''are'', then one would have reason to object that it does no good to represent a group and its elements in terms of their actions on the group elements themselves, since one still does not know what the latter entities ''are''.  Notice that the form of this objection is reminiscent of a dilemma that is often thought to obstruct the beginning of an inquiry into inquiry.  A similar pattern of knots occurs when one tries to explain the process of formalization in terms of its effects on the term ''formalization''.  In each case, the resolution of the difficulty turns on recognizing a distinction between the active and passive modes of existence that go with each nameable objective.

These reservations are crucial to understanding the form of explanation that a regular representation provides, that is, what it explains and what it does not.  If one is seeking an ontological explanation of what a group and its elements "are", then one would have reason to object that it does no good to represent a group and its elements in terms of their actions on the group elements themselves, since one still does not know what the latter entities "are".  Notice that the form of this objection is reminiscent of a dilemma that is often thought to obstruct the beginning of an inquiry into inquiry.  A similar pattern of knots occurs when one tries to explain the process of formalization in terms of its effects on the term "formalization".  In each case, the resolution of the difficulty turns on recognizing a distinction between the active and passive modes of existence that go with each nameable objective.

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 4, 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 4.  Both are abelian (commutative), but one is cyclic and the other is not.  The cyclic group on 4 elements is commonly referred to as "Z4".  (The German words "Zahl" = "number" and "Zyklus" = "cycle" together make the notation "Zn" suggestive of the integers mod n, which form a cyclic group of order n under the addition operation.)  The acyclic group on 4 elements is usually called the "Klein 4 group" and notated as "V4".  (The German name "Vierbein" is the substantive form of an adjective that means "four legged".)

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;.)

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.

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.