Let $H, K\leq G$. I was wondering what you call the "product" $HK$ of $H$ and $K$.

I was trying to verbalise the steps of showing $G$ is a semidirect product:

  • Normality of $H$: $H\unlhd G$.

  • Trivial intersection: $H\cap K=1$.

  • Product: $HK=G$.

However, I feel that there has to be a better word than "product" here.

Is there a "correct" answer? If so, I would appreciate it if you were to tell me what this answer is...

  • $\begingroup$ I've always called it a product of subgroups; Wikipedia seems to agree with me. $\endgroup$ – Clive Newstead Nov 29 '12 at 14:04
  • $\begingroup$ The phrase "product of subgroups" appears nowhere in this article... $\endgroup$ – user1729 Nov 29 '12 at 14:38
  • $\begingroup$ Wikipedia silently agrees with Clive. $\endgroup$ – Phira Nov 29 '12 at 22:02
  • $\begingroup$ @user1729: But what is a group subset which is a group? It's a subgroup... $\endgroup$ – Clive Newstead Nov 29 '12 at 23:17
  • $\begingroup$ Yeah, okay. I think subgroup product it is then! $\endgroup$ – user1729 Nov 30 '12 at 9:59

In general, $\,HK\,$ could properly be called a thing, or simply a set.

Now, $\,HK\,$ is a subgroup itself iff $\,HK=KH\,$ , and this happens for example when at least one of the subgroups is normal, as in your case.

So you can really call $\,HK\,$ "the product of $\,H\,,\,K\,$ , which is a subgroup."

  • $\begingroup$ I suppose what my question is really asking is "$HK$ is the $X$-product of $H$ and $K$. What is $X$?". The fact that $HK$ is a subgroup is implicit (because $HK=G$). $\endgroup$ – user1729 Nov 29 '12 at 12:37
  • $\begingroup$ (Also, I am looking for a word or phrase to replace "Product" in my list. "Thing" and "Set" don't really fit the bill...) $\endgroup$ – user1729 Nov 29 '12 at 12:39
  • $\begingroup$ I don't think there is anything better than "product", but who knows? Perhaps I'm wrong...:) $\endgroup$ – DonAntonio Nov 29 '12 at 12:40
  • $\begingroup$ I have "set product" pencilled in. Is that offensively wrong, or a valiant stab? I don't want to use product on its own because of the ambiguity which could arise between it and "semidirect product". When you write something noone reads it properly: they read what they want to read, what they think is there. (Okay, almost noone...finitely many people read what you wrote properly...) $\endgroup$ – user1729 Nov 29 '12 at 12:47
  • $\begingroup$ Well, then perhaps you could first explain why $\,HK\,$ is in fact a subgroup, and then call $\,HK\,$ simply "the group" (generated by $\,H,K\,$, say, or the subgroup product...) . OTOH, this is so basic stuff that I think it is very unlikely anyone with some little experience would have any problem. $\endgroup$ – DonAntonio Nov 29 '12 at 12:51

$HK$ is called the complex product of $H$ and $K$.

Generally, any subset is called a complex in an older fashion (see for example this note), and their elementwise product was called the complex product.

  • $\begingroup$ Isn't it just called the complex product because $H$ And $K$ are complexes? Here, $H$ and $K$ are subgroups so we would just have the subgroup product, no? (Which I quite like and has just been pencilled in...) $\endgroup$ – user1729 Nov 29 '12 at 13:40
  • $\begingroup$ @Berci: I've not seen such this name before, even in a old book (like D.Gorenshtin's). Thanks any way for the note. $\endgroup$ – mrs Nov 29 '12 at 14:29
  • $\begingroup$ I think, this name 'complex' is dying off, because has too many other meanings. Actually, in Hungarian we have studied it by this word, and I'm sure it existed in English too.. $\endgroup$ – Berci Nov 29 '12 at 14:36
  • $\begingroup$ It is also known in German as Komplexprodukt. $\endgroup$ – Hagen von Eitzen Nov 30 '12 at 20:24

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.