74
votes
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 ...
- 6,722
42
votes
Accepted
Structure of groups of order $pq$, where $p,q$ are distinct primes.
For a general group of order $p$ and $q$, there are very few possibilities (though you need Sylow theorems to know this). The fact is, for $p>q$ and $G$ a group of order $pq$, we must have
$$G\cong ...
- 11.9k
22
votes
Automorphism group of direct product of groups
Ideally an automorphism $\phi$ of $H\times H$ would look like $(\begin{smallmatrix}\alpha & \beta \\ \gamma & \delta \end{smallmatrix})$, with
$$\begin{pmatrix}\alpha & \beta \\ \gamma &...
- 149k
20
votes
Accepted
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 ...
- 377k
13
votes
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 ...
- 10.3k
12
votes
Accepted
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, ...
- 50.9k
11
votes
Accepted
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.
...
- 423
11
votes
Accepted
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.
- 416k
11
votes
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 ...
- 68.8k
10
votes
Accepted
In the category $\mathbf{Set}$ is "the product of an empty set of sets a one-element set"?
What the exercise is saying is the following: let $\mathcal{C}$ be an empty family of sets (awkward, but $\mathcal{C}$ is just isomorphic to $\emptyset$). Then, $\prod\,\mathcal{C}=\prod\limits_{C\in\...
- 49.1k
9
votes
Accepted
In Kunneth Formula for Cohomology, the finitely generated condition is necessary.
There is a natural isomorphism $$\prod_{I\times J}\mathbb{Z}\cong\operatorname{Hom}\left(\bigoplus_I\mathbb{Z},\prod_J\mathbb{Z}\right),$$ since Hom turns direct sums in the domain coordinate into ...
- 316k
9
votes
Accepted
A finite group $G$ and fixed $k\geq 1$ where for every $n\geq 1$, the $n$-direct product $G^n=G\times\dots\times G$ is $k$-generated?
No. For given $k$ and sufficiently large $n$, the projections of all $k$ putative generators would be equal on two different components, and so they could not generate the full direct product.
To be ...
- 84.7k
9
votes
Accepted
When must the subgroup of a product be the product of subgroups?
To spill the beans: this happens if and only if both $G_1$ and $G_2$ are torsion groups, and for every $x\in G_1$ and $y\in G_2$, $\gcd(|x|,|y|)=1$.
Once you figure out the statement one can prove it ...
- 377k
8
votes
Structure of groups of order $pq$, where $p,q$ are distinct primes.
Consider $S_3$, the group of symmetries of the triangle. It's not cyclic and has order $6=2\cdot 3$.
- 359
8
votes
Concerning Groups having the property that intersection of any two non-trivial subgroups is non-trivial
You are asking for groups in which every non-trivial subgroup is essential.
We can classify all abelian groups with this property, as follows.
Embed $G$ in its injective envelope $D$, which is a ...
- 29.8k
8
votes
Accepted
Symmetric Direct Product Distributive?
It doesn't work because you are taking the symmetric outer product between things which aren't the same. By direct computation with
$$\chi_3(\hat R)=\frac16\left\{\chi^3(\hat R)+2\chi(\hat R^3)\pm3\...
- 15.8k
8
votes
Accepted
Does anything special happen when you replace the direct product in the definition for a wreath product with a central product?
If $Z \le Z(A)$ then you can define the analogue of the wreath product in which the copies $Z_\omega$ of $Z$ in $A_\omega$ are amalgamated, and the result is the quotient of the ordinary wreath ...
- 84.7k
8
votes
Accepted
If $G,H$ are finite groups, then $G\times G\cong H \times H$ implies $G \cong H$
$\newcommand{\Hom}{\text{Hom}}\newcommand{\Surj}{\text{Surj}}$I disagree with Arturo's comment: Proving this lemma is much easier than proving the Krull-Schmidt theorem! Most of the work is already in ...
- 59.9k
7
votes
Accepted
In GAP, How can I check whether a given group is a direct product?
There is the command StructureDescription(G) which gives a name for the group $G$ and could tell you whether $G$ is a direct product or not. However, it just gives ...
- 24.4k
7
votes
Is finite group isomorphic with the direct prouct of the Sylow $p$ subgroups?
Others already explained that you don't get a direct product of groups for that needs the elements from distinct subgroups to commute (or both to be normal). You won't get even a semi-direct product ...
- 128k
7
votes
The only group $G$ with one $A$ and one $B$ as composition factors is $G = A\times B$ (where $A$ and $B$ are non-abelian, finite and simple)
The answer to the question is yes, the only group with two nonabelian finite simple groups $A$ and $B$ as composition factors is the direct product $A \times B$.
The well-known Schreier Conjecture ...
- 84.7k
7
votes
Accepted
Are these two infinite groups isomorphic?
Every element of order $5$ in $G_2$ is itself five times another element. But this isn’t the case for $G_1$.
- 31.4k
7
votes
Accepted
If $\displaystyle \bigoplus_{i=1}^{n} \mathbb{Z} \cong \bigoplus_{i=1}^{m} \mathbb{Z}$ as groups, then $n=m.$
The idea of your proof is fine, but there are some issues. (Contrary to the comments, it is not "OK".)
The main issue is with your function $2\varphi: 2G\rightarrow\bigoplus_{i=1}^{n} \...
- 29.2k
7
votes
Accepted
Isomorphisms for Infinite Direct Products of Groups
Lets start with multiple ways of defining the Cartesian product:
$G_1\times G_2$ is defined as the set of all pairs $(g_1,g_2)$, where $g_i\in G_i$. Then $G_1\times\cdots\times G_n$ is defined ...
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6
votes
Is there a slick way to test whether $\Bbb Z_{mn}\cong \Bbb Z_m\oplus \Bbb Z_n$?
Yes, there is a nice rule that says
$$
\mathbb{Z}_{n}\oplus \mathbb{Z}_m \simeq \mathbb{Z}_{mn}
$$
if and only if $\gcd(m,n) = 1$.
Also, the external direct product $\oplus$ is associative, so $A\...
- 42.6k
6
votes
How $(\mathbb R,+)$ is a decomposible group?
We can see $\mathbb{R}$ as a vector space over $\mathbb{Q}$ (so using rationals as scalars). This has a base (using the axiom of choice, there is no explicit description for such a base!) $B = \{x_i : ...
- 235k
6
votes
Concerning Groups having the property that intersection of any two non-trivial subgroups is non-trivial
The groups ${\mathbb Z}_{p^{\infty}}$ with $p$ prime have this property, and there infinitely many of these - one for each prime. In fact they have a unique minimal nontrivial subgroup, which is ...
- 84.7k
6
votes
Example of a group $G$ such that $G = N_1\cdots N_n$ and $N_i \cap N_j = \{e\}$ for all $i \neq j$ but $G$ is not the internal direct product of them.
Let $H$ be a group with nontrivial center $Z$ and identity element $e$. Let $G=\{(a,b)\in H^2\mid a\equiv b\pmod{Z}\}$. Let $D = \{(h,h)\in H^2\mid h\in H\}$ be the diagonal subgroup of $H^2$.
The ...
- 12.3k
6
votes
In GAP, How can I check whether a given group is a direct product?
The question would be more interesting if you asked how you might do this rather than asking whether there is a command to do it in GAP.
Here is a first suggestion, which you could easily implements ...
- 84.7k
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