You're not about to guess what 'commutative group' is from it's name any more than you're going to guess what an 'abelian group' is, unless you have already learned what commutative or abelian means in the specific context of algebra.
I'm always baffled that programmers, who freely adopt frameworks and languages with mostly meaningless names (which incidentally makes them easier to google for), continue to insist that names are the major stumbling block in learning math. It's utter nonsense. You pick up the names and terminology quickly enough (or look them up if you forget); the hard part is applying them.
Sure if you're actively working in that subfield. I took abstract algebra in university and remember we treated groups and commutative groups, but I can't tell you what exactly rings were or what Abelian referred to. (But just like knowing the names of things is not the same as knowing things, forgetting these labels is not the same as forgetting the materials.)
Naming things is important. Yes it just takes one googling, but in many cases you are interfacing with many different topics and just reading straight up the useful descriptive name would allow keeping the flow without having to look up stuff.
Maybe this is more of a stumbling block for some than others. Often I find I have to make up mnemonics and other strategies like "the longer word is the one that ..." or "put the two terms in alphabetical order to match their descriptive names' alphabetical order".
Again, this doesn't happen for terms we use every day. But if you use it every year or so, it's a stumbling block to have to ask "which one was that again?" when the descriptive name would immediately clear it up.
I think it's mostly vanity and a "respecting the elders" and credit assignment thing that so many things are named for mathematicians instead of descriptive names. Having something named after you is like one of the biggest "awards" a mathematician can get. But this has no concern for didactics.
I doubt many lawmen would be able to infer the meaning of “communicative” from today’s meaning of “to commute”. https://www.merriam-webster.com/dictionary/commute defines it as ”change, alter”, “convert”, or “compensate”, or as a synonym for “to commutate”, with meaning “to reverse every other half cycle of (an alternating current) so as to form a direct current” (first used, according to that dictionary in 1890; https://projecteuclid.org/download/pdf_1/euclid.rmjm/1181070... claims Kronecker introduced the term “Abelian group” in 1870)
Unless dictionaries 150-ish years ago had significantly closer descriptions of the term, “Abelian”, being a new term not loaded with pre-existing definition might be the better choice for naming this property.
Does it? I think that could describe “associative” equally well as “commutative”.
Edit: “symmetric” might be an alternative for Abelian because the Cayley table (https://en.wikipedia.org/wiki/Cayley_table) of an Abelian group is symmetric. I’m not sure that’s immediately clear enough for laymen, though.
"Symmetric group" is already taken. Given a set S, the symmetric group G_s is the group of all permutations of S. Not to be confused with a permutation group of S, which is a group of some permutations of S.
I suppose we could rename "symmetric group" to "maximal permutation group", but stringing adjectives together is not a sustainable strategy for naming. Plus, it is not clear to me how accurate "maximal permutation group" is as a name. That is, the symmetric group S_3 is the maximal permutation group of {1,2,3}. However, it is not generally a maximal permutation group, as it is contained withing the permutation group S_4.
Symmetric is also good but it has a very distinct geometric interpretation. If we can avoid overloading terminology in a confusing way when designing these the names we should do so.
Perhaps pair-independent is a distinctive name for associative...
Well, introducing the new term “Abelian” certainly avoids overloading terminology in a confusing way :-)
“Pair invariant” IMO, isn’t good, certainly worse than “Pairing invariant”. “Parentheses invariant” might work, but of course would get just as confusing/incorrect as “pair invariant” once one moves from groups to fields.
