# Using presentations, prove $\frac{\Bbb{Z} \times \Bbb{Z}}{\langle(3,3)\rangle}\cong\Bbb{Z} \times \Bbb{Z_3}$.

## The Details:

In considering this question on the same isomorphism and trying to come up with an alternative proof of my own (than the one composed of the work in the question and @DerekHolt's comment), I got stuck.

I want to use the following presentation of $$\Bbb Z\times \Bbb Z$$,

$$\langle a,b\mid ab=ba \rangle,\tag{\mathcal{P}}$$

by killing some element of the presentation.

My guess is to let $$c=ab$$ then kill $$c^3$$ in $$(\mathcal{P})$$, since, say, $$(1,0)\mapsto a$$ & $$(0,1)\mapsto b$$ and $$a$$ & $$b$$ commute, some other pithy Tietze transformations might elicit an isomorphism of the quotient of $$(\mathcal{P})$$ by $$\langle (3,3)\rangle$$ with

$$\langle x,y\mid y^3, xy=yx\rangle,\tag{\mathcal{Q}}$$

a presentation of $$\Bbb Z\times\Bbb Z_3.$$

## The Question:

Using presentations, prove $$\frac{\Bbb{Z} \times \Bbb{Z}}{\langle(3,3)\rangle}\cong\Bbb{Z} \times \Bbb{Z_3}$$.

## Thoughts:

I really think I ought to be able to do this myself. I work with presentations an awful lot. However, it has taken me the better part of an hour to articulate my hunch.

• What about sending $\overline{(1,0)} \to x$ (rather $\overline{a} \to x$) $\overline{(1,1)} \to y$ (or rather $\overline{ab} \to y$)? You just need to show this is an isomorphism from your quotient group to $\mathbb{Z}\times\mathbb{Z}_{3}$. Sep 8, 2019 at 22:29
• This is probably a stupid question, @MorganRodgers, but what do you mean by, say, $\overline{a}$? It reminds me of tranversals à la Magnus et al., but I doubt that's what you mean. Sep 8, 2019 at 22:41
• I've corrected the notation to be clearer, @MorganRodgers; I still don't know what you mean by $\overline{a}$ and $\overline{ab}$. The overline is used differently in Magnus et al.'s book "Combinatorial Group Theory: [. . .]". Sep 9, 2019 at 1:05
• I mean if $g$ is in the group, $\overline{g}$ is the corresponding element in the quotient group (ie the coset $g + \langle (3,3)\rangle$) Sep 9, 2019 at 4:14
• No, $(\mathcal{P})$ and $(\mathcal{Q})$ are presentations. I'm using the tag{$\mathcal{X}$) code for each, @MorganRodgers. It's a standard tool here. Sep 9, 2019 at 8:17
You can introduce a new generator, $$c=ab$$, and then $$b$$ is redundant (since $$b = a^{-1}c$$). Then the subgroup we are quotienting by is just $$\langle c^{3} \rangle$$, so the quotient group becomes $$\langle a,c \mid c^{3},\ ac = ca \rangle$$ which is clearly isomorphic to the presentation $$(\mathcal{Q})$$.
In terms of the presentation (using Tietze transformations), we have \begin{align} (\mathcal{P}) &\to \langle a,b,c \mid ab=ba, c=ab\rangle \\ &\to \langle a,c \mid c=a^{-1} c a, c=c\rangle \\ &\to \langle a,c \mid ac=ca\rangle. \end{align}