# Tagged Questions

Should be used with the (group-theory) tag. A group $(G,*)$ is said to be abelian if $a*b=b*a$ for all $a,b\in G.$

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### Is $h^*$ injective in this case?

Let $N$ and $N'$ be finite rank free $\mathbb Z$-modules. Let $M=\operatorname{Hom}_{\mathbb Z \text{-mod}}(N,\mathbb Z)$ and $M'=\operatorname{Hom}_{\mathbb Z \text{-mod}}(N',\mathbb Z)$ . Suppose ...
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### Why $G=HK$ and $H\cap K=\{e\} \implies G = H\times K$?

Given that $G$ is finite abelian, in order to show that $G = H\times K$ one only needs to show that $G=HK$ and $H\cap K=\{e\}$. What motivates this and why is it only true for finite abelian groups?
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### The number of nonzero ring homomorphisms $\mathbb Z_{30}\rightarrow \mathbb Z_{42}$ [duplicate]

I have managed to prove the the number of group homomorphisms is $\mathbb Z_m\rightarrow \mathbb Z_n$ is $\gcd (m,n)$, which is my case is $6$. However, I was told that the number of nonzero ...
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### Let $G$ be an Abelian group with $|G| = n$. Let $p$ be prime with $p | n$. Show that the Sylow p-subgroup of $G$ consists of $e$ and ..

Let $G$ be an Abelian group with $|G| = n$ and let $p$ be prime with $p | n$. Show that the Sylow p-subgroup of $G$ consists of $e$ and all elements whose order is a power of $p$. Answer: By ...
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### Why is $\mathbb{Z}[1/p]$ the direct limit of $\mathbb{Z}\xrightarrow{p}\mathbb{Z}\xrightarrow{p}\mathbb{Z}\to…$?

This is an example from Algebraic Topology, by Hatcher. As far as I understand, I have to take the direct sum of all the $G_i$s (in this case, $\mathbb{Z}\oplus\mathbb{Z}\oplus...$) and quotient out ...
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### Why is Grp not an Abelian Category?

As I understand it, the category of groups (not just abelian groups) satisfies all of the definitions of an abelian category. It has all kernels/cokernels as well as products/coproducts. Further the ...
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### Relation of Smith normal form to basis of subgroup

Let $A$ be a finite abelian group of rank $2$. Let $\left\{ e_{1},e_{2}\right\}$ be a basis of $A$ and let $C=\left\langle 2e_{1}+3e_{2},2e_{1}+6e_{2}\right\rangle$ be a subgroup. (a) Find ...
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### Exercise 6.79 from Rotman's Advanced Modern Algebra

If $G$ is a nonzero abelian group show that $$\operatorname{Hom}_{\Bbb Z}(G,\frac{\Bbb Q}{\Bbb Z}) \neq \{0\}.$$
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### Group of units of a non-integral quotient ring

I would to like to know which product of cyclic groups the group $A^\times$ of units of the quotient ring $$A = \mathbb F_5[X] / ((X^2-2)^2)$$ is isomorphic to. I know that $A$ is not an integral ...
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### Proving that a group is abelian [closed]

Suppose we have a group $G$ with $|G| = 10$. How do I prove that if its center $Z$ is nontrivial, then $G$ is abelian? Thanks.
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### Isomorphism and $(\mathbb Q,+)$

Prove that $(\mathbb Q,+)$ is not isomorphic to $(H,+) \neq (\mathbb Q,+)$, a proper subgroup of $(\mathbb Q,+)$. $\mathbb Q$ is the rationals. I thought about taking the contradiction ...
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### Open mapping theorem for normed abelian groups

A norm on an abelian group is a function valued in $\mathbb{R}_{\geq 0}$ which satisfies $|x|=0 \Leftrightarrow x=0$, $|{-}x|=|x|$, and $|x+y| \leq |x|+|y|$, not necessarily $|z x| = |z| |x|$ for ...
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### Is there a non abelian group that characterize a one dimensional lattice structure?

Of all groups that characterize a one dimensional lattice structure (symmetry operations including translation, $C_2$, mirror plane, inversion point), is there a non abelian one? Moreover, Can it have ...