As @JohnAbbott recently point out, it is somewhat confusing that we have MatElem and MatRingElem on the one hand, but MatrixGroupElem on the other hand.
Even more so given that there is also MatrixElem which is an alias for Union{MatElem,MatRingElem}.
Now in the grand scheme of things, this is not very important. But I guess we could rename the group types? Keeping the old names as aliases of course for backwards compatibility.