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In Artin's Algebra, I get this:

If $A$ and $B$ are subsets of a group $G$, then

$$AB=\{ab\mid a\in A,\space\space b\in B\}$$

We will call this the product of the two subsets of the group, though in other contexts the term product may stand for the set $A\times B$

My question: Is there any specific name of this set $AB$? If so, then what is it?

I know $A\times B$ is the product of the two sets(specially it is called Cartesian Product). But in group theory, $AB$ comes frequently, we can't just call it product too!

Added: I found $A\times B$ is also called Direct product of groups

Thanks in advance.

Community
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MAN-MADE
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  • It is referred to as the set-theoretic product. – Nicky Hekster Aug 08 '17 at 06:13
  • @NickyHekster I googled set-theoretic product. all it was saying about Cartesian Product.... but my question is different. – MAN-MADE Aug 08 '17 at 06:16
  • But it is called product, you are not going to change that, as there is a product defined for the elements of the group. In my lectures I always used set-theoretic, to emphasize that $AB$ is merely a set and does not have to be a subgroup, although $A$ and $B$ can be subgroups. – Nicky Hekster Aug 08 '17 at 06:37
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    In German text books it is referred as complex product. – Wuestenfux Aug 08 '17 at 06:57
  • Sometimes "Frobenius product" (see https://en.wikipedia.org/wiki/Product_of_group_subsets). – Hans Lundmark Aug 08 '17 at 08:50

2 Answers2

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I don't know if there's a widely used term for that, but "Finite-dimensional Spaces" by W. Noll seem to call it member-wise product which should cause no confusion. That term ought not be misunderstood.

Another alternative is using the term "non-cartesian product". That way the reader will know that it's not the cartesian product at least.

Wikipedia is calling it "product of group subsets", but there's a risk of misunderstanding in such terminology. Wikipedia also uses the term "product of subsets", but the factors in cartesian product are also subsets. Your source seem to point out that in other contexts that term may refer to cartesian product.

skyking
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Probably there is no specific name of $AB$ other than product of groups.But the concept of $AB$ and $A \times B$ are not same.$A \times B$ is called internal direct product of $A$ and $B$.$AB=A \times B$ if $A$ and $B$ are normal subgroups of $AB$ and $A \cap B=\phi$ so $AB \ne A \times B$ in general.

Supriyo Halder
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