Let $K$ be the group generated by four elements $x_1,\cdots,x_4$ with relations that any simple commutator with repeated generator is trivial; for example, $[[x_2,[x_1,x_3]],x_3]=1$. As I have asked here, I am trying to use GAP to do some calculation with $K$. Now one problem is that there are too many relations. For example, all following are relations in $K$:
- $[[[[x_{\sigma(1)},x_{\sigma(2)}],x_{\sigma(3)}],x_{\sigma(4)}],x_i]=1$, where $\sigma\in S_4$, the symmetric group on four letters $\{1,2,3,4\}$ and $i\in\{1,2,3,4\}$.
- $[[[x_{\sigma(1)},x_{\sigma(2)}],x_{\sigma(3)}],x_i]=1$, where $\sigma\in S_3$, the symmetric group on three letters $\{1,2,3\}$ and $i\in\{1,2,3\}$. And by replacing $\{1,2,3\}$ by other three-element subsets of $\{1,2,3,4\}$, there are some more relations.
- $[[x_{\sigma(1)},x_{\sigma(2)}],x_i]=1$, where $\sigma\in S_2$, the symmetric group on two letters $\{1,2\}$ and $i\in\{1,2\}$. And by replacing $\{1,2\}$ by other two-element subsets of $\{1,2,3,4\}$, there are some more relations.
Now my question is:
- Is there any way that I can reduce the number of relations? I believe that some relations here must be redundant.
- Is it pratical to use GAP to do calculation with $K$? I am worried that too many defining relations may make the calculation impractical.