Determining group structure after computing homology: $\langle b,c \mid 2(b+c)=0, b+c=c+b \rangle.$

Question

I am trying to determine what group this is a presentation of: $$\langle b,c \mid 2(b+c)=0, b+c=c+b \rangle.$$ I am pretty sure it is $\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$, but am stuck on how to show it.

Answer 1

Put multiplicatively, your presentation is

$$\langle b,c\mid (bc)^2, bc=cb\rangle.$$

By Tietze transformations, introduce $a=bc$, so that $c=b^{-1}a$, giving the presentation

$$\langle a,b\mid a^2, a=(b^{-1}a)b\rangle,$$

which simplifies to

$$\langle a,b\mid a^2, ba=ab\rangle;$$

and this presentation yields, multiplicatively, the group $\Bbb Z_2\times \Bbb Z$. This is isomorphic to $\Bbb Z\oplus \Bbb Z/2\Bbb Z$.