Convoluted definition of a set: H(S)={x∈G∣∃n∈N,∃{x1,x2,⋯,xn}⊆S∪S−1,x=x1⋯xn}.

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skullsxtest7xt · Accepted Answer

No. H(S) is the set of all elements g of G such that g can be expressed as a finite product of elements of S∪S−1.  Maybe lets have a look at example. Consider G={0,1,2,3} with addition modulo 4 as group operation, i.e. G=Z4. Now let S={2}. Since we are dealing with addition, then S−1={2} because 2+2=0 in our group. Therefore S∪S−1={2}.  Next H(S) is the set of all elements g of G such that g=x1+⋯+xn for some x1,⋯,xn∈S∪S−1. Since our last set has exactly one element then only possibilities for g are: 2,2+2,2+2+2,  and which we can write in a compact way as n⋅2 for some n∈N. By removing duplicates (remember that 2+2=0) this gives us  H(S)={0,2}

Convoluted definition of a set: H(S)=\{x \in G| \exists n

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