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Groups

2.b) Consider G= , x*y be the fractional part of x+y .(i.e:x*y=x+y-[x+y] where [a] is the greatest integer less than or equal than a )
Construct a group H such that for each there exists an element of order , but non of the other orders are present.(Hint : use a subqroup of ):

I want to claim the group is H= under addition.
Although, each element in H had of order .
But for p, k belong to p+k won't belong to it.
Also the inverse does not belong to H

Please find my mistake for me. Thanks.

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Your idea is correct, but your mistake is that you fix the numerator to be 1.
Thus for example, (1/8)*(1/4)=3/8, which is not in your group H.
Here is my solution.
H = {k/2^n: n>=1, 0<=k<2^n} is the desired group.
In a detailed form,
H = {0, 1/2, 1/4, 3/4, 1/8, 3/8, 5/8, 7/8, ... }
Proof:
First, ...

Solution Summary

Groups are investigated.

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