# Terminology: $H$ and $K$ are subgroups. What is $HK$ called?

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...

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

• 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 $HK=G$). – user1729 Nov 29 '12 at 12:37
• (Also, I am looking for a word or phrase to replace "Product" in my list. "Thing" and "Set" don't really fit the bill...) – user1729 Nov 29 '12 at 12:39
• I don't think there is anything better than "product", but who knows? Perhaps I'm wrong...:) – DonAntonio Nov 29 '12 at 12:40
• 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...) – user1729 Nov 29 '12 at 12:47
• 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. – 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.

• 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...) – user1729 Nov 29 '12 at 13:40
• @Berci: I've not seen such this name before, even in a old book (like D.Gorenshtin's). Thanks any way for the note. – mrs Nov 29 '12 at 14:29
• 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.. – Berci Nov 29 '12 at 14:36
• It is also known in German as Komplexprodukt. – Hagen von Eitzen Nov 30 '12 at 20:24