# 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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### Struggling to understand example of Ideal which is not finitely generated

I'm working through an algebraic number theory book, but I can't understand the example shown below: I follow the example up till it assumes that $\frac{p_1}{q_1},...,\frac{p_n}{q_n}$ are the ...
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### Two term free resolution of an abelian group.

This is probably a very easy question but I think I am missing some background regarding free abelian groups to answer it for myself. In Hatcher's Algebraic Topology, the idea of a free resolution is ...
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### Difficulty understanding subgroups for certain simple groups

Firstly, I am studying at the high-school level so please excuse my lack of understanding of these concepts. Consider the group $\Bbb C_{3v}$ of symmetries of an equilateral triangular lamina. It ...
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### Extending a homomorphism from a subgroup to whole group where the target is not a divisible group

I was reading this post of stack exchange. So in the question if the circle group is replaced by $\mu_{p-1}$ which is the group of $(p-1)^{th}$ root of unity and if the group $G/H$ is assumed to a ...
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### Number of group homomorphism from $Z_8$ ⊕ $Z_2$ to $Z_4$ ⊕ $Z_4$

I know that there does not exist an isomorphism from $Z_8$ ⊕ $Z_2$ to $Z_4$ ⊕ $Z_4$ as there exist an element of order 8 in $Z_8$ ⊕ $Z_2$ and no element of order 8 in $Z_4$ ⊕ $Z_4$. But what about ...
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### Proving that a homomorphism is abelian [duplicate]

The question states... Let $\phi: G \to G$ be a homomorphism with the map $\phi(g)=g^2$. Prove $\phi$ is abelian. So far I have: Let $g$ and $h$ be in $G$. Then $\phi(g)=g^2$ and $\phi(h)=h^2$. ...
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### Order of elements in certain generating sets of non-abelian groups !!!

The following example is just to clarify the idea. Example: The dihedral group has the following presentation $$D_{2i}=\left<s,r/s^2=r^i=e,sr=r^{-1}s \right>$$ Let $S_1=\{s,r\}$, ...
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### Subgroups of abelian-by-finite groups

I am trying to prove that a subgroup of a abelian-by-finite group is also abelian-by-finite. I am not sure if I can use the same procedure that is used for a subgroup of a ...
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### $f:A\to B$ epimorphism if and only if $B/f(A)$ is a torsion module [closed]

I want to prove the double implication: $f: A\to B$ is an epimorphism in the category of torsion-free abelian groups $\Longleftrightarrow$ $B/f(A)$ is a torsion $\mathbb Z$-module. Thanks.
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### Divisible group $G\neq 0$ is not free

How do I show that a divisible group $G\neq 0$ is not free? I know that divisible means that for all elements $g$ in an abelian group $G$ and $n\in\mathbb{N}$ there is an element $a\in G$ such that ...
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### Structure theorems for infinitely generated Abelian groups

The classification theorem for finitely generated Abelian groups is well known and plays big role in mathematics. Are there any structure theorems about infinitely generated Abelian groups known?
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### Isomorphism between $A$ and $B$ if $A/kA$ and $B/kB$ have same order? [closed]

Let $A$ and $B$ be finitely generated abelian groups, where for all $k\geq1$ the orders of $A/kA$ and $B/kB$ are equal. Are $A$ and $B$ necessarily isomorphic? Why (not)?
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### Surjective endomorphism of abelian group is isomorphism

Let $A$ be a finitely generated abelian group and $f:A\rightarrow A$ a surjective homomorphism. How do I prove that $f$ is an isomorphism? And if $f$ were injective instead of surjective would the ...
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### Finding an explicit isomorphism from $\mathbb{Z}^{4}/H$ to $\mathbb{Z} \oplus \mathbb{Z}/18\mathbb{Z}$

There was a past qualifying exam problem, I was having trouble with, it is stated below as follows: In the group $G= \mathbb{Z} \times \mathbb{Z}\times \mathbb{Z}\times \mathbb{Z}=\mathbb{Z}^{4}$, ...
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### Proving property of cyclic groups

A user asked the following question. It was closed as off-topic, or rather as missing context, but it seems the context close reason doesn't exist, so off-topic was chosen. Here it is: I am having ...
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### Exercises Involving Torsion and Abelian Groups

I am working on the following group-theory exercises but I'm a little confused by how to begin proving them. Let $G$ be a group. Call $g \in G$ a $torsion \ element$ if $g$ has finite order $g^k = e$ ...
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### o(a)=m ,o (b) =n , ab=ba then o(ab)=lcm(m,n). What happens when b is the inverse of a?

Let $G$ be a group and let $a,b \in G$ s.t $O(a)=m$ and $O(b)=n$ and $ab=ba$. Then $O(ab)=lcm( m,n)$. My attempt: since $ab=ba$ then $HK=KH$ $|HK|=O(H)O(K)/O(H \cup K)$ $l=(mn)/O(H \cap K)$ ...
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### Why is abelianness such a precious property?

My abstract algebra teacher said the other day that constructions like ideals and cosets and normal subgroups are "trying to capture a little bit of abelianness." He has used phrases like "magic ...
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### Help with Abelian group and homomorphisms

Let A be an abelian group and k∈Z. a) Show that hk:A→A defined by hk (a)=ka is a homomorphism. b) In the case A= Z , show that any homomorphism Z→Z must be hk for some k . Any help ...
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### Let $(G, *)$ be a group and let $\{g,h\}$ be a subset of $G$. Prove that $(g*h)^{-1}=h^{-1}*g^{-1 }$.

Let $(G, *)$ be a group and let $\{g,h\}$ be a subset of $G$. Prove that $(g*h)^{-1}=h^{-1}*g^{-1}$. I know that I should show that $X*Y=Y*X=e$. But I don't know how to calculate it.
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### Subquotient of abelian group equivalent to “quotientsub” of abelian group?

Suppose we have an abelian group $G$. Suppose also that we have a "subquotient" $H$, which is a subgroup of a quotient group of $G$. If $H$ can be constructed in this way, when is it also true that ...
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### Group G with $\left(\forall g \in G: g^2=1_G\right) \implies \text{G is abelian}$ [duplicate]

I have the following task: Be $G$ a group with $\forall g \in G: g^2=1_G$ Prove that $G$ is abelian. I proved it this way: $\forall g \in G: g^2=1_G$ implies $\forall g \in G: g=g^{-1}$ ...
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### Let G be an abelian group, and let a∈G. For n≥1,let G[n;a] := {x∈G:x^n =a}. Show that G[n; a] is either empty or equal to αG[n] := {αg : g ∈ G[n]}… [closed]

We were given questions to study for our exam coming up. We have not covered much of this topic, so any help would be greatly appreciated! Let $G$ be an abelian group, and let $a\in G$. For $n≥1$, ...
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### On ordered abelian groups containing $\mathbb{Z}$

Let $\Delta$ be an ordered abelian group containing $\mathbb{Z}$ as a subgroup of index $e$. I need to show that for any positive element $\delta \in \Delta$, we have $e\delta \geq 1$. I have no ...
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### Commutativity of a Lie algebra $\Rightarrow$ the Lie group is abelian

Let $G$ be a Lie group, $\mathfrak{g}$ it's Lie algebra. Assume $[x,y]=0 \, \, \forall x,y \in \mathfrak{g}$. Is it true that $G$ is abelian? Remarks: (1) The other direction ($G$ abelian ...
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### $H=\{x^2:x\in G\}$ then $H\unlhd G$

We have $H<G$ and element of $H$ is of the form $x^2$ where $x \in G$. H is a normal subgroup of $G$. The factor group $G/H$ is abelian. I tried first one by showing that $gh^2g^{-1} \in H$. ...
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### Find the order of a subgroup of $\mathbb C^\times$

Let $\mathbb{C}^\times$ the multiplicative group of complex numbers different of zero. Let $H$ the subgroup $\mathbb{C}^\times$ of generated by $\{i, e^{\frac{2i \pi}{5}}, -1\}$. Find the order of ...
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Let the group $\Bbb{Z_2} = \{e, a\}$. We are given the quotient group $\Bbb{Z_2} / \{e\}$. So this gives us a set of left cosets: $\Bbb{Z_2} / \{e\} = \{e\{e\}, a\{e\}\} = \{\{e\}, \{a\}\} \neq \{e, ... 0answers 50 views ### Number of isomorphism classes of abelian groups of any order Let$N$be the order of an abelian group. The prime factorization is given by$N=\prod_{i=1}^{n}p_{i}^{e_{i}}$with$p_{1}< p_{2}< \dots <p_{n}$and$e_{i}\geq 1$. Let$\pi(n)$denotes the ... 4answers 305 views ### Prove that a group is cyclic [closed]$G$is abelian of order$35$. and for all$x\in G$,$x^{35}=e$. I need to show that$G$is cyclic. This seems perfectly obvious but I dont know how to write the proof. Help would be appreciated! ... 2answers 69 views ### How can you tell whether two groups are homomorphic/isomorphic? [closed] Suppose you have two groups,$G$and$H$. I've been taught the following definitions: "$G$is homomorphic to$H$iff there exists some function$\theta$which gives the mapping$\theta : G ...
According to page 158 of Dummit and Foote's Abstract Algebra (3rd edition): Theorem. (Fundamental Theorem of Finitely Generated Abelian Groups) Let $G$ be a finitely generated abelian group. Then ...