Find an isomorphism from the group G= {1, i, -1, -i}, to themultiplicative group

Question

Find an isomorphism from the group G= {1, i, -1, -i}, to themultiplicative group H= {1 0    i 0     -i0    -1 0         0 1,   0-i ,   0 i ,   0 -1 Find an isomorphism from the group G= {1, i, -1, -i}, to themultiplicative group H= {1 0    i 0     -i0    -1 0         0 1,   0-i ,   0 i ,   0 -1

Explanation / Answer

You can look at the 4x4 table of multiplications and rearrange theelements to make the same table: there are a few ways to do it, for example, you can match theelements of G to the a_1,1 element in the matrix in H 1 -> 1 0         0 1 i-> i 0      0 -i -1 -> -1 0            0-1 -i -> -i 0          0 i (having the same multiplicative table for finite groups is a validextensive proof for isomorphism) btw, a nice way to look at this group is as operations (complexmultiplication) of turning counter clockwise(i)/clockwise(-i)/flipon both x and y axis/unit and just place the elements on the diagram

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