# Normal closure in groups

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

In general, what are the standard approaches to compute the normal closure of a subset of a finitely presented group? Are there algorithms?

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In general, the normal closure will not be finitely generated. In what sense do you want to "know" it? – Mariano Suárez-Alvarez Oct 9 '10 at 16:12
It is f.g. here. – user641 Oct 9 '10 at 16:59
Well, the question is in general... – Mariano Suárez-Alvarez Oct 9 '10 at 17:38
@Mariano: If you know many senses, please let me know. – Max Black Oct 10 '10 at 14:04

You can compute the normal closure by computing the quotient, and then considering the kernel of the quotient homomorphism.

For the example you gave, let $N$ be the normal closure of $y$ in $G$. Then $G/N$ has presentation $$\langle x,y \mid x^{12}y = yx^{18},y=1\rangle$$ This presentation reduces to $\langle x \mid x^{12} = x^{18}\rangle$, which is the same as $\langle x \mid x^6 = 1\rangle$.

Thus $G/N$ is a cyclic group of order 6, and $N$ is the kernel of the homomorphism $G\to G/N$. In particular, the normal closure of $y$ consists of all words for which the total power of $x$ is a multiple of 6.

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Nice, thank you. – Max Black Oct 10 '10 at 14:20