Jacob Stein

2022-02-15

Convoluted definition of a set: $H\left(S\right)=\left\{x\in G\mid \mathrm{\exists }n\in N,\mathrm{\exists }\left\{{x}_{1},{x}_{2},\cdots ,{x}_{n}\right\}\subseteq S\cup {S}^{-1},x={x}_{1}\cdots {x}_{n}\right\}$.

skullsxtest7xt

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 $S\cup {S}^{-1}$.
Maybe lets have a look at example. Consider $G=\left\{0,1,2,3\right\}$ with addition modulo 4 as group operation, i.e. $G={\mathbb{Z}}_{4}$. Now let $S=\left\{2\right\}$. Since we are dealing with addition, then in our group. Therefore $S\cup {S}^{-1}=\left\{2\right\}$.
Next H(S) is the set of all elements g of G such that $g={x}_{1}+\cdots +{x}_{n}$ for some ${x}_{1},\cdots ,{x}_{n}\in S\cup {S}^{-1}$. 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 $n\cdot 2$ for some $n\in \mathbb{N}$. By removing duplicates (remember that $2+2=0$) this gives us
$H\left(S\right)=\left\{0,2\right\}$

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