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

Jacob Stein

Jacob Stein

Answered question


Convoluted definition of a set: H(S)={xGnN,{x1,x2,,xn}SS1,x=x1xn}.

Answer & Explanation



Beginner2022-02-16Added 15 answers

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 SS1.
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 S1={2} because 2+2=0 in our group. Therefore SS1={2}.
Next H(S) is the set of all elements g of G such that g=x1++xn for some x1,,xnSS1. 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 n2 for some nN. By removing duplicates (remember that 2+2=0) this gives us

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