What are the elements of $\Bbb Z\oplus \Bbb Z_2$ and $\Bbb Z∗\Bbb Z_2?$

Question

What are the elements of the following groups?

(1) $\Bbb Z\oplus \Bbb Z_2$ ,

(2) $\Bbb Z∗\Bbb Z_2$

Thanks in advance.

Answer 1

The elements of $\Bbb Z\oplus\Bbb Z_2$ are pairs $(z, [w]_2)$ for $z\in\Bbb Z$ and $[w]_2\in\Bbb Z_2$, whereas since $$\langle a, b\mid b^2\rangle$$ is a presentation for $G=\Bbb Z\ast\Bbb Z_2$, the elements of $G$ are of the form $\prod_{i=0}^na^{\alpha_i}b^{\varepsilon_i}$, where $n\in\Bbb N\cup\{0\}$, $\alpha_i\in \Bbb Z\setminus\{0\}$ (with $\alpha_0$ potentially $0$), and $\varepsilon_i=1$ (with $\varepsilon_n$ potentially $0$).