Let G = {1, a, b, c} be the Klein 4-group and following the example below, label

Question

Let G = {1, a, b, c} be the Klein 4-group and following the example below, label the elements a,b,c,d as 1,2,3,4. The example below shows that the left multiplication by the group element a gives rises to the permutation (12)(34).

a * 1 = a1 = a and so sigmaa(1) = 2
a * a = aa = 1 and so sigmaa(2) = 1
a * b = ab = c and so sigmaa(3) = 4
a * c = ac = b and so sigmaa(4) = 3

Mirroring the calculations for a, describe how the left multiplication of element b acts on the set 1,2,3,4.

Let G = {1, a, b, c} be the Klein 4-group and following the example below, label the elements a,b,c,d as 1,2,3,4. The example below shows that the left multiplication by the group element a gives rises to the permutation (12)(34). Mirroring the calculations for a, describe how the left multiplication of element b acts on the set 1,2,3,4.

Explanation / Answer

b * 1 = b1 = b so sigmab(1) = 3 ( for b is the 3rd element)
b * a = ba = ab = c so sigmab(2) = 4 (for c is the 4th element)
b * b = bb = 1 so sigmab(3) = 1 (for 1 is the 1st element)
b * c = a so sigmab(4) = 2 (for a is the 2nd element)
so the permutaion is 1 2 3 4
3 4 1 2 so it can be represented as sigmab = (1,3)(2,4)

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