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An '''isomorphism''' is a homomorphism that is one to one and onto, or bijective.  Systems that have an isomorphism between them are called '''isomorphic''' to each other and belong to the same '''isomorphism class'''.  From an abstract point of view, isomorphic systems are tantamount to the same mathematical object, differing at most in their manner of presentation and the details of their representation.  Usually these differences are regarded as purely notational, a mere change of names.  Thus, they are seen as accidental or accessory features of the object, corresponding to different ways of grasping the objective structure that is the main interest of the study but not considered as essential parts of its ultimate constitution or even necessary to its final comprehension.

An "isomorphism" is a homomorphism that is one to one and onto, or bijective.  Systems that have an isomorphism between them are called "isomorphic" to each other and belong to the same "isomorphism class".  From an abstract point of view, isomorphic systems are tantamount to the same mathematical object, differing at most in their manner of presentation and the details of their representation.  Usually these differences are regarded as purely notational, a mere change of names.  Thus, they are seen as accidental or accessory features of the object, corresponding to different ways of grasping the objective structure that is the main interest of the study but not considered as essential parts of its ultimate constitution or even necessary to its final comprehension.

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 an "addition table" or as a "multiplication 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 "*", as in x*y = z.  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 "." 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 an "addition table" or as a "multiplication 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 "*", as in x*y = z.  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 "." or by concatenation) appear in the same context, then the star is retained for the group operation.