Multiplication group theory
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Multiplication group theory

[From: ] [author: ] [Date: 12-01-29] [Hit: ]
which shows that [xw][u] = [xw][zy] = [(xw)(zy)] = [1+ (k + r + krn)n] = [1],as desired.......

(technically, we should probably write [x],[w] and [x][w] = [xw] = [1],

to avoid confusion with actual integers, but this is rarely done in practice).

we also have z in Zn, with wz = 1.

so we need to find some u in Zn so that (xw)u = 1.

the obvious candidate is zy, so let's see if that works:

(xw)(zy) = x(wz)y = x(1)y = xy = 1.

but let's look at this "on the integer level":

[x][y] = [1] means [xy] = [1], that is: xy = 1 + kn (for some integer k).

[w][z] = [1] means wz = 1 + rn, for some integer r.

now we want to show that there is some u such that [xw][u] = [1].

the u we choose is u = zy.

now [xw][zy] = [(xw)(zy)] = [x(wz)y].

remember, wz = 1 + rn, so

x(wz)y = x(1 + rn)y = (xy)(1 + rn)

(since ordinary multiplication of integers is commutative)

= (1 + kn)(1 + rn) = 1 + (k + r + krn)n,

which shows that [xw][u] = [xw][zy] = [(xw)(zy)] = [1 + (k + r + krn)n] = [1],

as desired.
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keywords: group,theory,Multiplication,Multiplication group theory
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