# Normal closure in teams

For instance, claim $G = \langle x , y \ | \ x^{12}y=yx^{18} \rangle$. I need to know what is the normal closure of $y$ in $G$.

As a whole, what are the typical strategies to calculate the regular closure of a part of a finitely offered team? Exist formulas?

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2019-05-18 23:08:03
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For the instance you offered, allow $N$ be the regular closure of $y$ in $G$. After that $G/N$ has discussion $$\langle x,y \mid x^{12}y = yx^{18},y=1\rangle$$ This discussion lowers to $\langle x \mid x^{12} = x^{18}\rangle$, which coincides as $\langle x \mid x^6 = 1\rangle$.
Hence $G/N$ is a cyclic team of order 6, and also $N$ is the bit of the homomorphism $G\to G/N$. Specifically, the regular closure of $y$ contains all words for which the complete power of $x$ is a numerous of 6.