Isomorphisms
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When structures are not isomorphic
Equations are omitted for technical reasons - download the original pdf
We can illustrate the difference of two structures by showing that there cannot be an isomorphism between them Consider the groups ¢4 and K ¢4 is the cyclic group of four elements. K is the Klein group and is identical to the symmetry group of a rectangle. The combination tables of ¢4 and K are [Diagram goes here - download the original to see it.] In K every element is its own inverse, but in ¢4 only 0 and 2 are their own inverses. Therefore there cannot be a structure preserving isomorphism. More formally, we show that every possible mapping from ¢4 to K does not satisfy the structure preserving property. [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] It would be possible to show a similar failure for every possible bijection from to K. When two groups are isomorphic then their combination tables are such that one differs from the other only in (1) the labels used to designate the elements; (2) the order in which the rows (and columns) are set down. Therefore, in order to show that two groups are isomorphic it is sufficient to rearrange the combination table for one of them - by permuting rows - to demonstrate that they are the same structure because the elements follow each other in the same order.
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Contents of
Isomorphisms
1 Isomorphisms
2 When structures are not isomorphic
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