[duetosymmetry]

> fixed-length binary strings are a group where the operation is xor (or '^') and inv(x) = x

Most people would call this group (Z/2Z)^n, i.e. n copies of Z/2Z under addition. Similarly, for "matrix addition" you could say Z^(mn) or R^(mn) for matrices with dimension mxn with elements Z or R (or whatever you are filling your matrices with). Your only non-abelian example is matrix multiplication, i.e. GL(n,R), general linear transformations of dimension n over R. It may or may not be interesting that this is mainly focused on abelian groups.