Note: My question is not directly about solving the below problem.
Let $G$ be a group with presentation given by $$G= \langle a,b,c \mid ab =c^2a^4,bc=ca^6,ac=ca^8,c^{2018}=b^{2019}\rangle.$$ Determine order of the group $G$.
In the above problem, we see that the group is defined by four equations.
My question is, how do we know if these equations are independent?
For instance, how could we know if $ab=c^2 a^4$ and $bc=ca^6$ implies $ac=ca^8$ or not? Is there some easy algorithim for this?
I think this is in some way related to inverted linear equation to write unknowns in term of coefficients, but I'm not sure how.
This is in general (in the nonabelian case) an extremely hard problem that cannot have a general solution in the Turing Machine model. See Word Problem or Dehn's Problems. This does not preclude some approaches working in particular cases.
In the abelian case it is solved by the Smith Normal Form.