# Questions tagged [direct-product]

For questions about the direct product of groups, rings, fields or categories. Use (group-theory), (ring-theory), (field-theory) or (category-theory).

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### Let $G$ be a finite group and $M$ be a maximal subgroup of $G$. If $G = Z(G)M$, then $M$ is normal in $G$

I need to prove that if $G$ is a group, $M$ is a maximal subgroup of $G$ and $Z(G) \nsubseteq M$,then $M \unlhd G$. Is true that $G = Z(G)M$, right? Is this enough?
1 vote
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### Let $K=\{(g,g):g \in G\}$ where $G$ is a group. There are no subgroups $H_1,H_2$ of $G$ such that $K=H_1\times H_2$

How to formally prove the following: Let $K=\{(g,g):g \text{ is a member of G}\}$ where $G$ is a group There are no sub-groups $H_1,H_2$ of $G$ such that $K=H_1 \times H_2$ If I think about it it's ...
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### Given groups $G_1, G_2$, if $K$ is a subgroup of $G_1 \times G_2$, are there $H_1\leq G_1$ and $H_2\leq G_2$ such that $K=H_1\times H_2$? [duplicate]

Duplicate Question Isn't a duplicate, I didn't mention they are finite. Question: Given groups $G_1, G_2$, if $K$ is a subgroup of $G_1 \times G_2$, are there $H_1\leq G_1$ and $H_2\leq G_2$ such ...
1 vote
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### Let $G,G'$ be finite groups of orders $m,n$ respectively. What is the order of $G×G'$? [duplicate]

Let $G,G'$ be finite groups of orders $m,n$ respectively. What is the order of $G×G'$? I have started studying serge lang's undergraduate algebra. This is the question from books group theory ...
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### Minimal normal subgroups of Product of simple groups

Known Result: Let $$G= S_1 \times S_2 \times\dots\times S_n,$$ where each $S_i$ are non-abelian simple groups. Then $S_i$'s are the minimal normal subgroup of $G$. (Even $S_i$'s are the only minimal ...
1 vote
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### Trying to understand the differences between $\mathbb{Z}_2 * \mathbb{Z}_2$ vs $\mathbb{Z}_2 \times \mathbb{Z}_2$

I’m trying to understand the differences between free products and direct products with an example: $\mathbb{Z}_2 * \mathbb{Z}_2$ vs $\mathbb{Z}_2 \times \mathbb{Z}_2$. If I understand correctly, the ...
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### Show that there is no subgroup B of $\mathbb Z ^2$ such that $\mathbb Z ^2 / B \cong \mathbb Z_3 \times \mathbb Z_6 \times \mathbb Z_{21}$

I have the following question in group theory: Show that there is no subgroup $B$ of $\mathbb Z ^2$ such that $\mathbb Z ^2 / B \cong \mathbb Z_3 \times \mathbb Z_6 \times \mathbb Z_{21}$ I don't ...
1 vote
39 views

### Two competing definitions of the direct sum of vector spaces

There seem to be two competing definitions of the direct sum of vector spaces. The first one characterises it as the same as the Cartesian product for a finite number of vector spaces, and for an ...
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### Show subgroup isomorphic to direct product

$T$ is a minimal generating set for $S_n$. $T=${$(i,i+1),i=1,...,n−1$}. For any fixed $x$ with $1≤x<n$, the set $W =$ $T$\ {$x,x+1$} does not generate $S_n$. $W$ generates some subgroup of $S_n$. ...
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### Does there exist some isomorphism from $Q_8$ to $\Bbb Z_4 \times\Bbb Z_2$?

I am wondering if some direct decomposition exists for quaternion group. I think that I am mixing some things, but let me explain and ask for clarification, tips from your side to let me understand my ...
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### A direct product $G=\mathbb{Z}_4\times\mathbb{Z}_4\times\mathbb{Z}_4\times\mathbb{Z}_4$ of the cyclic group $\mathbb{Z}_4$ of order $4$.

Let $G=\mathbb{Z}_4\times\mathbb{Z}_4\times\mathbb{Z}_4\times\mathbb{Z}_4=\mathbb{Z}_4^4$ be a direct product of $4$ copies of the the cyclic group $\mathbb{Z}_4$ of order $4$. Can one regards $G$ as ...
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### Two exercises by Robinson on supersolvable groups seem to contradict.

This is concerning (part of) Exercise 5.4.5 and Exercise 5.4.6 of Robinson's, "A Course in the Theory of Groups (Second Edition)". I have done the first one; the second might take me a while....
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### If $HK$ is a subgroup but not equal to $G$ (where $H,K<G$), then are $H$ and $K$ normal in $HK$?

I have asked a very similar question already and many have answered it also. If $HK$ is a subgroup of $G$ (where $H$ and $K$ are subgroups of $G$), then are $H$ and $K$ normal in $HK$? But all the ...
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### Define a binary operation on {e,(12)}×{e,(123),(132)} so that it becomes isomorphic to $S_3$ [closed]

Since {e,(12)} isomorphic to $Z_2$ and {e,(123),(132)} isomorphic to $Z_3$ and gcd(2,3)=1, so, {e,(12)}×{e,(123),(132)} isomorphic to $Z_6$ which is not isomorphic to $S_3$. But can a binary ...
1 vote
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### Is the given condition necessary for $HK$ to be isomorphic to $H\times K$?

$H \triangleleft G,K \triangleleft G, H \cap K=e$, then, $HK$ is a subgroup of $G$ and $HK$ isomorphic to $H\times K$, where, $\triangleleft$ denotes normal subgroup. If $G$ isomorphic to $H\times K$,...
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### For $P$ the set of all primes, ${\rm Hom}(\Bbb Z,\sum_{p\in P}\Bbb Z_p)\not\cong\prod_{p\in P}{\rm Hom}(\Bbb Z,\Bbb Z_p)$

${\rm Hom}(\mathbb Z, \sum_{p\in P}\mathbb Z_p)$ and $\prod_{p\in P}{\rm Hom}(\mathbb Z, \mathbb Z_p)$ are not isomorphic where $P$ is the set of all primes. I was checking the elements of each to see ...
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### Under what conditions can we say that $G/N \cong S \implies G \cong N \times S$?

For a recent project (which I have since completed) I needed to derive the automorphism group of the cube graph, and I wanted to do so with some reasonable degree of rigor. I defined a group action of ...
1 vote
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### Count the subgroups of order $p^2$ in $\mathbb{Z}_{p^3}\oplus\mathbb{Z}_{p^2}$

How many subgroups of $\mathbb{Z}_{p^3}\oplus\mathbb{Z}_{p^2}$ are there with order equal to $p^2$? My attempt: Let $H$ be a subgroup of $\mathbb{Z}_{p^3}\oplus\mathbb{Z}_{p^2}$ and $|H|=p^2$, then ...
1 vote
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### Isomorphisms for Infinite Direct Products of Groups

Here is a question from Section 2.13 of Herstein's "Topics in Algebra" (2nd edition): If $G_{1}$, $G_{2}$, $G_{3}$ are groups, prove that $(G_{1} \times G_{2}) \times G_{3}$ is isomorphic ...
1 vote
The four homomorphisms created by the direct product construction are each an example of a more general construction of homomorphisms involving groups $G$, $H$ and $G\times H$. By using the same ...