I am not sure about a suitable name for the type of group that I am going to discuss and in the title I have used the word "Brunnian", which was borrowed from "Brunnian links" in topology. Recall that a Brunnian link is a nontrivial link that becomes trivial if any component is removed.
Now I am considering a type of groups with similar properties: A group $G$ of $n$ gengrators $x_1,\cdots,x_n$ such that in $G$, for any fixed $1\leq i\leq n$, if any element $y$ is can be expressed as a product of $x_1,\cdots,\hat{x_i},\cdots,x_n$ (that is, there is an expression of $y$ which contains no powers of $x_i$), then $y$ is equal to the identity element in $G$.
I just wonder if there is any known results for this type of groups. Has anyone done research with these groups before? A particular question that I am interested is to find out the cardinality of the center of a "Brunnian type" group, or at least give an upper bound of the order of the $p$-torsion component of the center.
Any comments and references will be greatly appreciated.
