Determine whether H is a subgroup of the complex numbers C with addition H={a+bi∣a,b∈R,ab≥0}

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Determine whether H is a subgroup of the complex numbers C with addition  H={a+bi∣a,b∈R,ab≥0}

d2saint0 · Accepted Answer

Clearly, the complex number 0=0+0i in H as a=b=0 and ab=0≥0.  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=0≥0  Hence, 1+0i∈H  Consider the complex number 0−i.  For the complex number 0−i, a=0, b=−1 and ab=0≥0.  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&amp;lt;0.  Hence, 1−i∉H.  Thus, 1+0i,0−i∈H, but their ∑1−i!nH.  Therefore, H is not closed under addition.  So, H is not a subgroup of C under addition.

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

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