Determine whether H is a subgroup of the complex numbers C with addition H = {a+bi|a,b in R, ab>=0}

Annette Arroyo 2020-11-08 Answered
Determine whether H is a subgroup of the complex numbers C with addition
H={a+bia,bR,ab0}
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d2saint0
Answered 2020-11-09 Author has 89 answers
Clearly, the complex number 0=0+0i in H as a=b=0 and ab=00.
So, H is a non empty subset of C.
In order for H to be a subgroup of C, H must be closed under addition.
Consider the complex number 1+0i.
For the complex number 1+0i, a=1, b=0 and ab=00
Hence, 1+0iH
Consider the complex number 0−i.
For the complex number 0−i, a=0, b=−1 and ab=00.
Hence, 0−i in H.
Now, (1+0i)+(0−i)=(1+0)+(0−1)i=1−i.
For the complex number 1−i, a=1, b=−1 and ab=−1<0.
Hence, 1iH.
Thus, 1+0i,0iH, but their 1i!nH.
Therefore, H is not closed under addition.
So, H is not a subgroup of C under addition.
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