# 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?

You can calculate the regular closure by calculating the ratio, and afterwards taking into consideration the bit of the quotient homomorphism.

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.

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