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Finally, to introduce two pieces of language that are often useful: an '''endomorphism''' is a homomorphism from a system into itself, while an '''automorphism''' is an isomorphism from a system onto itself.
Finally, to introduce two pieces of language that are often useful: an '''endomorphism''' is a homomorphism from a system into itself, while an '''automorphism''' is an isomorphism from a system onto itself.
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 32.1 illustrates the general scheme of a group operation table. In this case the group operation, treated as a “multiplication”, 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.
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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.
