This is Exercise 1.1.5(c) of "Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations," by Magnus et al.
The Question:
Give a presentation of $C_4$ using one generating symbol $b$ and two defining relators, neither of which is $b^4$ or $b^{-4}$.
My Attempt:
We have $$C_4=\langle b\mid b^2=b^{-2}, b^6=b^2\rangle,$$ but this doesn't seem to be in the spirit of the question.
Please help :)
Is this correct?
Here $b^4=b^{12}b^{-8}=1\cdot 1=1$ is a defining relator and so $$\langle b\mid b^8, b^{12}\rangle$$ is the presentation since $\langle b\mid b^4\rangle$ is a presentation.