This question is related to [Can I derive $i^2 \neq 1$ from a presentation $\langle i, j \mid i^4 = j^4 = 1, ij = j^3 i\rangle$ of Quaternion group $Q$?](https://math.stackexchange.com/questions/424991/can-i-derive-i2-neq-1-from-a-presentation-langle-i-j-mid-i4-j4-1/425017?noredirect=1#425017)

I know I'm going too far but let me just ask...

1) Is indeed $\langle i \mid i^4 =1\rangle$ a presentation for the cyclic group of order 4?

2) If so, then do we just assume that $i^2 =1$ or should we prove that $i^2 =1$ cannot be derived from $i^4 =1$, i.e., the unprovability of $i^2 =1$ from $i^4=1$?

For the below questions I need answers only if the latter is the case in the above question 2.

3) Then how do we rigorously prove the unprovability?

4) As for 
$$\langle i, j \mid i^4 = j^4 = 1, ij = j^3 i\rangle = Q
$$
(Quaternion group), how do I prove unprovability of $i^2 =1$ from the given presentation? Or how do I even guess that $i^2 \neq 1$?