This post is way off topic, and a bit in the weeds, and maybe even a bit silly, which is why I’m throwing it here rather than getting back to making my proper blog work. I really will do that, but… ok, not today.
I’ve been trying to learn some abstract algebra. I never studied it in school, and sometimes I bump into people describing operations in “group-speak” (throwing around words like “abelian” and “symmetric group”), and I’d like to try to keep up with what they’re saying even if I probably will never do an actual proof in anger.
Anyway, I got myself confused about the meaning of “order” when talking about groups vs elements. It’s the kind of dumb “how could you not know this at this point in the course?” question that you’re embarrassed to ask (and so I blog about it because I have an under-developed sense of shame). So… let’s ask some LLMs! And also search engines. And how would I figure this out?
Good news: it took longer to write this up than to fix my misunderstanding. OK news: nothing lied to me (well, much). Bad news: well, we’ll get to that as it comes up.
Here’s the exact text I stuck into a bunch of things:
Is the order of a group the lcm of the order of its elements?
To be clear to anyone who knows abstract algebra: I was confused and thought this might be the definition of order. I was misunderstanding the point of Lagrange’s theorem. To anyone who doesn’t know abstract algebra, the correct answer is just “the number of elements in the group.” It really is that simple, and I was making it really over-complicated.
Kagi has become my go-to search engine. I like the model of “I pay you, and you provide a service, that’s it, that’s the whole business model.” It gave me several links that weren’t particularly useful (basically ones that prove Lagrange’s theorem, which assume you already know the answer to my question).
But the third link was to a Saylor Academy text book, which answered my question:
In group theory, a branch of mathematics, the term order is used in two closely-related senses:
• The order of a group is its cardinality, i.e., the number of its elements.
OK, that’s cleared up.
Am I done? Of course not! I had to dig through three or four links, most of which confused me more, before I finally found the link to an open-source textbook, where I could go double-check the definition. The internet should be better than that!
Other search engines
To quickly summarize other search engines (because you’re all familiar with those, and of course the point is to get to LLMs):
Duck Duck Go – (OK) Second link was Wikipedia, which gives the answer in the first sentence. For completeness, searching for the same question on Wikipedia directly also gets you right to the page.
Google – (Bad) The Q&A section gives several links to different questions on Stack Exchange, Quora, and Studysmarter.us (?). Importantly, the second answer Google highlights is excerpted as: “Yes, in fact, the order of an element in a finite group…” (The “…” is in the Google excerpt.) You have to carefully notice that this is the answer to a different question. If you read quickly, you could easily walk away thinking “yes” is the answer. This is a big problem with search engine excerpting IMO. If you scroll down past the Q&A section, the first (unrelated) math.stackexchange answer, the “People also ask” section, and then two more links, you will finally come to the Wikipedia article that does answer the question and is where I should have started anyway.
No. For instance if G = Z2 ⊕ Z2, then | G | = 4, but the least common multiple of the order of it’s group elements is 2.
Bing added the bold on “No.” This is the only search that directly answered my question, with citation, in one step. The Math.SE link is a bit funny, though. Five years ago, someone had almost exactly the same question as I did. It was answered in a couple of useful ways, had an accepted answer, and then was closed as off-topic. (!?!?!) This question does not show up in most web searches (other than Bing) and does not show up (at least not prominently) when I put my search directly into math.se.
Bing definitely won this one. Also, of course, just going to Wikipedia.
The LLMs - Bing
OK, but we all know what this was really about. How do the LLMs handle this kind of question? Well, let’s start with Bing, since it answered the question so well through search.
Me: is the order of a group the lcm of the order of its elements?
Bing: No, the order of a group is not necessarily the least common multiple of the order of its elements. For instance, if G = Z2 ⊕ Z2, then |G| = 4, but the least common multiple of the order of its group elements is 2.
Precisely the same answer as in the “quick answer” from search, except this time it’s plagiarism. It is word-for-word the answer from Aweygan on math.se, but gives no link, citation, or even hint that this isn’t original content.
Maybe I could get the citation by asking?
Me: What’s the source of your answer?
Bing: My answer is based on my internal knowledge and information. I am programmed to provide accurate and reliable information to the best of my ability.
And of course I must check ChatGPT:
No, the order of a group is not necessarily the least common multiple (lcm) of the orders of its elements.
The order of a group is the number of elements in the group. On the other hand, the order of an element in a group is the smallest positive integer n such that the element raised to the nth power is the identity element of the group.
Hey, that’s actually really good. Just the right about of extra information. I’d like citations of course, but just being right about basic facts without plagiarism is pretty good (after all, the math.SE answer doesn’t provide citations, either). I like this. This is good.
And it goes on:
It is true that the order of an element divides the order of the group, by Lagrange’s theorem. However, the lcm of the orders of the elements may be larger than the order of the group. For example, consider the cyclic group of order 6 generated by a. The group contains elements of orders 1, 2, 3, and 6. The lcm of these orders is 6, but the order of the group itself is only 6.
Therefore, the order of a group cannot be expressed in terms of the lcm of the orders of its elements, in general.
Ah, a counter-example where the lcm is larger than the order. Very helpful. In the case where the order is six and the lcm is…six?
Oh ChatGPT. If only you’d quit while you were ahead.