# Tag Info

### Direct Sum vs. Direct Product vs. Tensor Product

I won't even attempt to be the most general with this answer, because I admit, I do not have a damn clue about what perverted algebraic sets admit tensor products, for example, so I will stick with ...
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### Is there another name for Goursat's Lemma on subgroups of a direct product of groups?

Goursat's Lemma is in several group theory textbooks, but not always by name. It appears in Marshal Hall's classic The Theory of Groups. In the AMS Chelsea Publications edition, it is Theorem 5.5.1 on ...
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### Structure of groups of order $pq$, where $p,q$ are distinct primes.

Let $\lvert G \rvert = pq$ for primes $p, q$ such that $q < p$ and $q \not \mid p-1$. Let $n_p$ and $n_q$ be the number of Sylow $p$-subgroups and Sylow $q$-subgroups, respectively. By the Third ...
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### Does there exist a group $G$, such that $G\otimes H \cong G$ for all finite groups, $H$?

For the first part of the question, the answer is yes, though it takes a slightly 'monstrous' construction. Since the finite groups are countable, we can enumerate them as $G_1$, $G_2$, $\ldots$; now, ...
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### If $H$ is a subgroup of a finite abelian group $G$, then $G$ has a subgroup that is isomorphic to $G/H$.

Since no one answered my question, I did some reading and found out that this is a very well known result. By using the concept of a "basis" of an abelian group, I did the following proof. ...
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### Why does the empty set not get a relation in a cartesian product?

Actually, $\emptyset$ is not element of $A$ (or of $B$). The set $\emptyset$ is a subset of $A$ (and of any other set), but that's irrelevant for your question.

### What theories are preserved under the product?

The definition of product you suggest is indeed the standard one. It can be naturally extended to define the product $\prod_{i\in I} M_i$ of any family of structures $(M_i)_{i\in I}$. Here I'll ...
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