Changes

where <math>{}^{\backprime\backprime} + {}^{\prime\prime}\!</math> denotes the operation in Table&nbsp;35.1.

where <math>{}^{\backprime\backprime} + {}^{\prime\prime}\!</math> denotes the operation in Table&nbsp;35.1.

Then the mapping <math>h : X_1 \to X_2\!</math> whose ordered pairs are given by:

Then the mapping h : X1 >X2 whose ordered pairs are given by:

h = {<1, 0>, <a, 1>, <b, 2>, <c, 3>}

| <math>h ~=~ \{ (1, 0), (a, 1), (b, 2), (c, 3) \}\!</math>

|}

constitutes an isomorphism from Z4(.) to Z4(+).

constitutes an isomorphism from <math>Z_4(\cdot)\!</math> to <math>Z_4(+).\!</math>

This fact can be verified in several ways:  (1) by checking that the map h is bijective and that h(x.y) = h(x) + h(y) for every x and y in Z4(.), (2) by noting that h transforms the whole multiplication table for Z4(.) into the whole addition table for Z4(+) in a one to one and onto fashion, or (3) by finding that both systems share some collection of properties that are definitive of the abstract group, for example, being cyclic of order 4.

This fact can be verified in several ways:  (1) by checking that the map h is bijective and that h(x.y) = h(x) + h(y) for every x and y in Z4(.), (2) by noting that h transforms the whole multiplication table for Z4(.) into the whole addition table for Z4(+) in a one to one and onto fashion, or (3) by finding that both systems share some collection of properties that are definitive of the abstract group, for example, being cyclic of order 4.