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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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    Yeah, okay. I think subgroup product it is then!2012-11-30

2 Answers 2

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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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    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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    It is also known in German as *Komplexprodukt*.2012-11-30