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Tagged with quotient-group infinite-groups
13 questions
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Subgroups and Quotient Groups of Infinite bounded Abelian p-groups
I am looking to a reference to a solution of Exercise 13 (page 91) of L. Fuchs, Infinite Abelian Groups, Vol. 1, Academic Press, 1970. There is no solution to this exercise in a collection Exercises ...
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Let $K\le H\le G$ with $[G:H]<\infty$. Is $[G:K]<\infty$ necessarily?
While reading this proof for the Tower Law for Subgroups, the data for the theorem only stipulates that H is a subgroup of G with finite index, and that K is a subgroup of H. Later in the proof ...
- 665
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The factor group $\mathbb{R} / \mathbb{Z}$ under addition has an infinite number of elements of order 4 [closed]
The factor group $\mathbb{R} / \mathbb{Z}$ under addition has an infinite number of elements of order 4.
The answer is false, but I cannot still understand it. I thought that it is true since there ...
user1168774
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Is there a group with a proper subgroup such that the quotient is isomorphic to itself? [duplicate]
While studying for my finals this question came across my mind. Is there a group $G$ and a proper subgroup $H$ such that $G/H\cong G$? I immediatly discarded a finite group or even the integers, as ...
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Must a virtually abelian group surject onto the integers?
Let $G$ be a finitely generated, infinite, virtually abelian group; that is,
$G$ is infinite, but
there is a finite set $X\subseteq G$ generating $G$, and
there exists abelian $H\leq G$ with $[G:H]&...
- 10.1k
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Which groups have non-trivial cyclic quotient groups?
Under what conditions can it be determined whether or not any group G has some subgroup N such that the quotient group G/N is a cyclic group? In other words, how can it be determined whether or not a ...
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For $H\le G$, abelian $G$, show that $r_0(H)+r_0(G/H)=r_0(G)$, where $r_0(K)$ is the torsionfree rank of $K$
This is Exercise 4.2.7(i) of Robinson's "A Course in the Theory of Groups (Second Edition)". According to Approach0, it is new to MSE.
I've asked two previous questions on based on earlier ...
- 47.2k
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Quotient of $\mathbb{R^*}$ by cyclic group product
For any $k \in \mathbb{R^*}, \langle k\rangle$ is a normal subgroup.
Consider the case $k \neq 1, k>0$. Then, via the surjective homomorphism $$\varphi: \mathbb{R^*} \to \mathbb{T}\times\{\pm1\}, \...
- 2,495
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Let $H, K \lhd G$, then $G = HK \iff G / (H \cap K) \cong G / H \times G / K$
This is a followup of this question
So, let $ (G,*) $ be a group.
Say $ H,K \lhd G $
If $ G $ is finite, then from the previously asked question we conclude: $ \ G = HK \iff G / (H \cap K) \cong G / H ...
- 363
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The order of elements in infinite quotient groups
There are two statements that my professor made today that I'm hoping I can get some more clarification on.
The first is that $\mathbb{Q}/\mathbb{Z}$ is an infinite quotient group where every element ...
- 203
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Properties of quotient group $\frac{\Bbb{Q}}{\Bbb{Z}}$
Consider the quotient group $\frac{\Bbb{Q}}{\Bbb{Z}}$
So $\frac{\Bbb{Q}}{\Bbb{Z}}$ = $\{\frac{p}{q}: p, q \in \Bbb{Z}, q \neq 0\}$
Now consider $\Bbb{Z}$+$\frac{r}{m}$ where $m$ is a natural number ...
- 669
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Finding the cosets of $\langle 360\Bbb{Z}, +\rangle$ in $\langle\Bbb{R}, +\rangle$.
I'm struggling to find the cosets of $\langle\mathbb{R}, +\rangle / \langle 360\mathbb{Z}, +\rangle$.
Usually I only find cosets of finite groups, in which case I know the cardinality of the subset ...
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Group Isomorphic to $(\mathbb{Z} \times \mathbb{Z})/ \langle(a,b),(c,d)\rangle$
I have come across an interesting question regarding quotient groups of $\mathbb{Z} \times \mathbb{Z}$:
Suppose $(a,b)$ and $(c,d)$ are two independent elements of $\mathbb{Z} \times \mathbb{Z}$. ...
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