Finite/Infinite groups Which of the following statements is/are true?
a) There are infinitely many finite groups in which every non-identity element have order $2$.
b)There exists an infinite group in which every non-identity element have order $2$.
c)There exists an infinite group in which there are elements of order $n$ for all $n\in \Bbb N$.
d)There are infinitely many infinite groups in which each non-identity element have finite order.
My attempt: Option 2 seems correct as I have an example of such a group( Power set of Natural numbers under the binary operation of symmetric difference is one such example) but I am unable to conclude anything about other options.
About option 1, it seems that it is true keeping in mind $\Bbb Z_2\times \Bbb Z_2\times\cdots$ but still I am confused. For the last two options, I don't have any conclusive idea which I can apply here. Please guide.
 A: (a) For $n\geq 1$, $\prod_{i=1}^n\Bbb{Z}_2$ is a finite group whose every nontrivial element has order $2$. So there are infinitely many such groups.
(b) $\prod_{i=1}^{\infty}\Bbb{Z}_2$ is the desired group.
(c) You may consider the example $\Bbb{Q}/\Bbb{Z}$. Then for every $n\in \Bbb{N}$, $\frac{1}{n}+\Bbb{Z}$ is element of order $n$ in $\Bbb{Q}/\Bbb{Z}$.
(d) For $n\geq 2$, $\prod_{i=1}^{\infty}\Bbb{Z}_n$ is an infinite group whose every nontrivial element has finite order. So there are infinitely many such groups.
A: If it's easier for you to conceptualize, you can take each element to be finite, even though the group as a whole is infinite.
For instance, for (2), we can take each element of $G$ to be a finite sequence consisting of $1$ and $-1$ and ending in $-1$ (the identity is the null sequence). To find $ab$, we multiply the values of $a$ and $b$ pairwise, then terminate the sequence at the last $-1$. If they are of different lengths, we append $1$ to the shorter one enough times to make them the same length, then multiply. $G$ is infinite because although each individual element has a finite length, there's an infinite number of those finite lengths.
A similar strategy works for (3) and (4). For (3), we take finite sequences in which the $n$th value is an element of $\mathbb Z_n$, unless the sequence is length 1, the last value isn't $1$. For (4), we simply take variations on (3). For instance, we can start the sequence at $\mathbb Z_k$. Or take $k$ copies of (3), etc.
