# Cardinality of a group with presentation

Let $$G$$ be a group with presentation $$G = \langle a,b,c|a^{\alpha} = b^{\beta}= c^{\gamma} = 1, [a,b] = c, [a,c] = [b,c] = 1 \rangle$$ Where $$\alpha$$, $$\beta$$ and $$\gamma$$ are natural numbers. I want to show that $$G$$ is finite and compute its cardinal.

First, it's easy to show that $$G = \{a^{r}b^{s}c^{t} |r,s,t \in \mathbb{Z}\}$$, obviously with non-uniqueness of $$r,s$$ and $$t$$. So then one tries to define a map (not necessarily a morphism, we are only interested in cardinality) $$\begin{array}{rcl} \varphi \colon \mathbb{Z}/\alpha\mathbb{Z}\times\mathbb{Z}/\beta\mathbb{Z}\times\mathbb{Z}/\gamma\mathbb{Z} & \longrightarrow & G\\ (r + \alpha\mathbb{Z}, s + \beta\mathbb{Z}, t + \gamma\mathbb{Z}) & \longmapsto & a^{r}b^{s}c^{t} \end{array}$$ This map is well-defined (thanks to the trivial commutators, the equation $$a^{r}b^{s}c^{t} = a^{r^{'}}b^{s^{'}}c^{t^{'}}$$ is equivalent to $$a^{r-r^{'}}b^{s-s^{'}}c^{t-t^{'}} = 1$$) and onto. The problem is the injectivity, and I suspect that to achieve this I'll need to change $$\gamma$$ by the order of $$c$$ in $$G$$. Any ideas?

I'll give a slightly different approach that I find more intuitive:

(Thanks Derek for pointing out a previous error)

We first use the relations to simplify the presentation:

$$a^b=ac$$ and $$a^c=a$$ implies $$a=a^{b^\beta}=ac^\beta$$ so $$o(c)\mid\beta$$.

Similarly $$(b^{-1})^a=cb^{-1}$$ and $$b^c=b$$ implies $$o(c)|\alpha$$.

Together with $$c^\gamma=1$$ this gives $$o(c)|\gcd(\alpha,\beta,\gamma)=\delta$$

Hence $$G=\langle a,b,c|a^\alpha=b^\beta=c^\delta=1,[a,b]=c,[a,c]=[b,c]=1\rangle$$.

But this is the presentation for $$(C_\alpha\times C_\delta)\rtimes C_\beta$$.

Hence $$|G|=\alpha\beta\gcd(\alpha,\beta,\gamma)$$

• Thank you, clearly answering questions too late, have corrected Oct 16, 2019 at 18:47
• OK I've deleted my comment, which was full of typos. Oct 16, 2019 at 19:22
• Thanks! Maybe this is too stupid, but I can't see why this new presentation is the presentation for the semidirect product $(C_\alpha\times C_\delta)\rtimes C_\beta$. Oct 16, 2019 at 20:18
• Not stupid at all! See for example here Oct 16, 2019 at 21:28
• But which morphism are you taking in the semidirect product? And why do you need to make the gcd appear in the presentation? Oct 18, 2019 at 19:07