# 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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### Is Aut(G×H) isomorphic to Aut(G) × Aut(H) [duplicate]

Please suggest me the proof.Am stuck with it.I saw somewhere that it will be true if (o(G),o(H))=1 but why?
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### Showing $\Bbb Z\times \Bbb Z$ is not a free group. [closed]

I need a little help with prove that $\Bbb Z\times\Bbb Z$ is not a free group by using the universal property, I thought of creating a map between a subset $S$ of $\Bbb Z\times\Bbb Z$ to a group of ...
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### 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)

Is it true that if $A$ and $B$ are two non-abelian finite simple groups, then the only finite group $G$ which has one copy of $A$ and one copy of $B$ as composition factors is $G = A \times B$? If not,...
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### Every abelian $p$-group is the direct product of cyclic groups.

Theorem $:$ Every abelian $p$-group is the direct product of cyclic groups. I have started reading that proof from this Proof Wiki article. Here I have understood everything before the element $b$ is ...
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### Find a nonabelian subgroup $T$ of $S_3 \times \Bbb Z_4$ of index $2$, generated by elements $x,y$ such that $|x|=6$, $x^3=y^2$, and $yx=x^{-1}y$.

Find a nonabelian subgroup $T$ of $S_3 \times \Bbb Z_4$ of index 2, generated by elements $x,y$ such that $|x|=6$, $x^3=y^2$, and $yx=x^{-1}y$. I gave it a try but still, don't reach the solution. ...
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### Find identity and inverses of the group $(\Bbb{Z}_6,+)\times(\Bbb{Z}_5,\cdot)$.

Let $(\Bbb{Z}_6,+)$ and $(\Bbb{Z}_5,\cdot)$ be two groups. Define an operation for $(\Bbb{Z}_6,+)\times(\Bbb{Z}_5,\cdot)$ such that $(\Bbb{Z}_6,+)\times(\Bbb{Z}_5,\cdot)$ is a group. Find the ...
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### Universal Property of Products in $A$-modules

I am studying some notes in which the second Universal Property of Products in $A$-modules is defined as following: Given an $A$-modules family $\{M_i\}_{i\in I}$, for all test $A$-module $T$ we ...
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### What's this product of groups?

Let $G=\Bbb Z[\frac12]/\Bbb Z$ be the (additive) Prufer 2-group and let $\Bbb Z$ be the group of integers. Let $X=G\times \Bbb Z$ be the following product: Let $g_n$ be the subgroups of $G$ in ...
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### Let $H =\mathbb{Z}/2\mathbb{Z}\times G$ I want to show that H is isomorphic to a subgroup of $G$ and vice-versa

I'm trying to solve a few exercises and I have no clue for this one: Let $G$ be a group formed by $(a_1,a_2,...)$, an infinite series of elements of $\mathbb Z/4\mathbb Z$. Show that $G$ is ...
### Find number of subgroups of order $p$ in $\mathbb{Z}_p^n$
Let $p$ be a prime. Find number of subgroups of order $p$ in $\mathbb{Z}_p^n = \mathbb{Z}_p\times\cdots\times\mathbb{Z}_p$, '$n$' times. Consider any element $g\ne 1 \in \mathbb{Z}_p^n$, then order ...
### Does $G\cong H\times K$ imply $H\unlhd G?$
I want to prove the following exercise. If a group $G$ is the direct product of subgroups $H,K$, then $K$ is isomorphic to $G/H$. To prove this, I think I need first to show $H$ is normal in $G$. ...