Questions tagged [quotient-group]

This tag is for questions relating to "Quotient Group".

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Quotient group of $\mathbb{C^*}$ by $n$th roots of unity

I need to find image and kernel of the following homomorphism of groups ($\mathbb{C^*}$ is multiplicative group of complex numbers) $f: \mathbb{C^*} \rightarrow \mathbb{C^*}$ given by $f(z)=z^n$. I ...
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Groups of order $16$ with a cyclic quotient of order $4$

Question: I am interested in (a) listing the groups $G$ of order $16$ which have a cyclic quotient of order $4$; (b) in each case knowing in how many essentially different ways this occurs (...
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Suppose that $H$ and $K$ are normal subgroups of $G$ such that $H \cap K=\{e\}$. Show that $G$ is isomorphic with some subgroup of $G/H \times G/K$. [duplicate]

Let $G$ be a group with identity $e$. Suppose that $H$ and $K$ are normal subgroups of $G$ such that $H \cap K=\{e\}$. Show that $G$ is isomorphic with some subgroup of $G/H \times G/K$. It seems ...
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To prove $O/aO$ is finitely generated

Let $R$ be one dimensional Noetherian integral domain and $K$ be it's fraction field. Let $L/K$ be a finite extension, and $O$ be the integral closure of $R$ in $L$. Then, why $O/aO$ ($a$ is an ...
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Correspondence theorem: Subgroups of $\mathbb Z^2$ that contain $2\mathbb Z \times 4\mathbb Z$

Determine, and identify, the subgroups of $G=\mathbb Z^2$ that contain the smaller subgroup $H=2\mathbb Z \times 4\mathbb Z$. Note: I would like to identify subgroups not just determine them like in ...
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finitely generated quotient group: From $G/H = \langle g_1H, \ldots, g_m H \rangle$, we have $G=\langle g_1, \ldots, g_m \rangle H$?

Please help me with my own proof for this: A group with finitely generated normal subgroup and finitely generated quotient is finitely generated itself Let $G$ be a group with $G \trianglerighteq H$ ...
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Computing quotient groups arising from Baer sum

My question is a general one about computing quotient groups of (a subgroup of) a direct sum of $\mathbb{Z}$'s and $\mathbb{Z}/n\mathbb{Z}$'s, but it arises from computing the Baer sum of two ...
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If $N$ and $H$ are normal subgroups of $G$, then $N\cap H$ is also normal in $G$ and $G/(N\cap H)$ is isomorphic to a subgroup of $(G/N)\times(G/H)$?

I've already shown that $N\cap H$ is a normal subgroup of $G$, so I just need to determine an isomorphism. My problem is that I'm not entirely sure I understand what $(G/N)\times(G/H)$ looks like. I ...
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Classification of $\mathbb{Z}\times \mathbb{Z} \times \mathbb{Z} / \langle (n,n,n) \rangle$

I want to classify the abelian groups of the form \begin{align} \mathbb{Z}\times \mathbb{Z} \times \mathbb{Z} / \langle (n,n,n) \rangle \end{align} using fundamental theorem of finitely generated ...
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In $F=F^{ab}(A)$, define $f\sim f'$ if and only if $f-f'=2g$ for some $g\in F$. Show $F/\sim$ is finite if and only if $A$ is finite.

this is a question from Aluffi's Algebra: Chapter 0, Exercise II.5.10. The full question is the following: Given a free abelian group over a set $A$, denoted $F=F^{ab}(A)$, we define an equivalence ...
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If $(g,a) \mapsto g \cdot a$ is an action of $G$ on $A$ and $H \vartriangleleft G$, is $(\overline{g},a) \mapsto g \cdot a$ an action of $G/H$ on $A$?

I'm learning about group actions from Dummit and Foote, and I have a question about the following remark that the authors make in section 4.1: In particular, an action of [a group] $G$ on [a nonempty ...
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The structure of $\Bbb Z\times\Bbb Z/H$, where $H=\langle(3,-2)\rangle$

Consider the group $\DeclareMathOperator{\bZ}{\mathbb{Z}}\DeclareMathOperator{\bN}{\mathbb{N}}G=\bZ\times\bZ/H$ where $H=\langle(3,-2)\rangle$ (in general it looks like it works the same for subgroups ...
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Let $G$ be abelian s.t. $r_p(G)<\infty$ if $p=0$ or a prime. Prove $G/nG$ is finite for all $n>0$.

This is Exercise 4.3.9(b) of Robinson's "A Course in the Theory of Groups (Second Edition)". According to Approach0, it is new to MSE. This question is referred to later on in the text. The ...
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Let $p$ be odd prime.Why $a,b∈\mathbb Z_p^×-{(\mathbb Z_p^×)}^2$ implies $a/b∈{(\mathbb Z_p^×)}^2$?

Let $p$ be odd prime.Why $a,b∈\mathbb Z_p^×-{(\mathbb Z_p^×)}^2$ implies $a/b∈{(\mathbb Z_p^×)}^2$ ? For example, $2, -1∈\mathbb Z_3^×-{(\mathbb Z_3^×)}^2$ and $-2∈{(\mathbb Z_3^×)}^2$. I heard the ...
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A homeomorphism to the open upper-half complex plane

I am reading a quick course on modular forms, and I do not understand a lemma: Let $\alpha = \begin{pmatrix}a&b\\c&d \end{pmatrix} \in SL_2(\mathbb{R}), z\in \mathbb{C}^{*}$ (the Riemann ...