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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...

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    I've always called it a product of subgroups; [Wikipedia](http://en.wikipedia.org/wiki/Product_of_group_subsets) seems to agree with me.2012-11-29
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    The phrase "product of subgroups" appears nowhere in this article...2012-11-29
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    Wikipedia silently agrees with Clive.2012-11-29
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    @user1729: But what is a group subset which is a group? It's a subgroup...2012-11-29
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    Yeah, okay. I think subgroup product it is then!2012-11-30

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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."

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    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 $HK=G$).2012-11-29
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    (Also, I am looking for a word or phrase to replace "Product" in my list. "Thing" and "Set" don't really fit the bill...)2012-11-29
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    I don't think there is anything better than "product", but who knows? Perhaps I'm wrong...:)2012-11-29
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    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...)2012-11-29
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    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.2012-11-29
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    I know, but I just want to be clear!2012-11-29
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    Well, unless somebody else pops up and gives a better idea, I think I can tell you confidently that you will be clear if you use any of the ideas above.2012-11-29
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$HK$ 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.

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    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...)2012-11-29
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    @Berci: I've not seen such this name before, even in a old book (like D.Gorenshtin's). Thanks any way for the *note*.2012-11-29
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    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..2012-11-29
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    It is also known in German as *Komplexprodukt*.2012-11-30