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The multiplicative and additive versions of what is abstractly the same group, <math>Z_4,\!</math> can be used to illustrate the concept of a group isomorphism.

The multiplicative and additive versions of what is abstractly the same group, <math>Z_4,\!</math> can be used to illustrate the concept of a group isomorphism.

Let the multiplicative version of <math>Z_4\!</math> be formalized as follows:

Let the multiplicative version of Z4 be formalized as:

Z4(.) = X1  = <X1, *1, e1>  = <{1, a, b, c}, ., 1>,

| <math>Z_4(\cdot) ~=~ \underline{X}_1 ~=~ (X_1, *_1, e_1) ~=~ ( \{1, a, b, c \}, \cdot, 1),\!</math>

where "." denotes the operation in Table 34.1.

where <math>{}^{\backprime\backprime} \cdot {}^{\prime\prime}\!</math> denotes the operation in Table&nbsp;34.1.

Let the additive version of Z4 be formalized as:

Let the additive version of <math>Z_4\!</math> be formalized as follows:

Z4(+) = X2  = <X2, *2, e2>  = <{0, 1, 2, 3}, +, 0>,

| <math>Z_4(+) ~=~ \underline{X}_2 ~=~ (X_2, *_2, e_2) ~=~ ( \{0, 1, 2, 3 \}, +, 0),\!</math>

where "+" denotes the operation in Table 35.1.

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

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

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