From Gallian, Contemporary Abstract Algebra:
...if G is a group and H is a subgroup of G, it is not always true that aH = Ha for all a in G. There are certain situations where this does hold, however, and these cases turn out to be of critical importance in the theory of groups. It was Galois, about 185 years ago, who first recognized that such subgroups were worthy of special attention.
A subgroup H of a group G is called a normal subgroup of G if aH = Ha for all a in G. We denote this by H $\triangleleft$ G.
Are subgroups with aH = Ha named normal ("usual, typical, or expected"), from they being excess in examples, and not being rare? Or is there any other reason? And is there any reason for choosing the notation $\triangleleft$?
I am asking these questions, as knowing the relation with names and notation, seems to eliminate keeping memory for the mapping of name and to what it corresponds to, via relation.