An abelian group is a group where each element commutes with any other element in the set, so $ab=ba$.
I was reading the wikipedia article on the anticommutative property. I think this could be generalised to arbitrary groups as follows:
Suppose $G$ embed into the group of units of some ring $R$ (so for $g\in G$ the element $-g$ makes sense, as $-1$ is a unit). Then $a, b\in G$ anticommute if $ab=-ba$.
Another, possibly more general, interpretation is:
Let $-1$ be a distinguished central element of order two in $G$ (so for $g\in G$ the element $-g=g(-1)=(-1)g$ makes sense). Say that $a, b\in G$ anticommute with respect to $-1$ if $ab=-ba$.
I was therefore wondering: Is there a specific name for a group where any two elements in the group either commute or anti-commute (with respect to a fixed element $-1$)? That is, for $a,b$ in $G$ either $ab=ba$ or $ab=-ba$.