Questions tagged [quotient-group]
This tag is for questions relating to "Quotient Group".
830
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Quotient group isomophic to $\mathbb{Z}$
In a certain document I am asked to show that if, given a group $G$, there is a normal subgroup $N$ such that $G/N \simeq \mathbb{Z}$, $\forall n \in \mathbb{N}$ there exists a subgroup $H$ such that $...
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For group $G,H$, $f: G \to H$ be an homomorphism, and $N$ be a normal subgroup of $G$, $\frac{G}{N} \cong Im(f) \implies N \cong Ker(f)?$. [duplicate]
I am a complete beginner to Abstract Algebra. Today I asked my professor the following:
For group $G$ and $H$, let $f: G \to H$ be an homomorphism, and $N$ be a normal subgroup of $G$, if $\frac{G}{N}...
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1
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Finding all the possible orders of elements in a quotient group
Let $A$ be an arbitrary Abelian group and let $H:=\{a\in A\mid\exists b\in A\mid a=b^3\}.$ Prove $H$ is a normal subgroup of $A$. Determine all the possible orders of elements in the group $A/H.$
My ...
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$G/H \cong \mathbb{Z}$ implies existence of $K \leq G$ s.t. $K \cap H = 1$ and $HK = G$
Let $H \lhd G$ and $G/H \cong \mathbb{Z}$. The claim is that there is a subgroup $K \leq G$ such that $K \cap H = 1$ and $HK = G$.
Is the following correct?
Take $a$ to be any element in a generator ...
- 7,970
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0
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36
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Stallings' Theorem
I am trying to understand Stallings' Theorem for lower central series. Here is the statement:
Say we have groups $A, B$ with lower central series $A=A_1, A_2, ...$ and $B=B_1, B_2, ...$ respectively. ...
- 1,074
2
votes
1
answer
99
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Quotient group by two isomorphic groups
In the processes of studying some questions I suddenly realized something basic but weird. We have $\mathbb{Z}\cong 2\mathbb{Z}$, but $\mathbb{Z}/\mathbb{Z}\ncong \mathbb{Z}/2\mathbb{Z}$. It looks ...
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Composition series with non isomorphic quotients
Question
Let $1\lhd G_1 \lhd \ldots \lhd G_n=G$ be a composition series of the group $G$. If for every $i\not= j$ the quotients $G_{i+1}/G_i$ and $G_{j+1}/G_j$ are non isomorphic, then show that every ...
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Openness of canonical projection of quotient topological group
I'm studying the topological groups, and have difficulties with understanding why for a canonical projection onto quotient $\rho$ for a subgroup $H$ a preimage of image of an open set $U$ $\rho^{-1}(\...
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Quotient of the group $\mathbb{Z}^2 \oplus \mathbb{Z} ^2 \oplus \mathbb{Z} ^2$
Does there exists $x_1$, $x_2$ , $y_1$ , $y_2 \in \mathbb{Z}^2$, such that
$$\frac{\mathbb{Z}^2 \oplus \mathbb{Z} ^2 \oplus \mathbb{Z} ^2 }{ \langle (x_1,y_1,0), (0,x_1,y_1) , (x_2,y_2,0), (0,x_2,y_2) ...
- 1,163
1
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1
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Subgroups of self dual groups
Let G be an infinite locally compact abelian group that is isomorphic to its own dual. If H is a closed subgroup of G, is it necessarily true that $H \cong \widehat{G/H}$?
I ask because in the case of ...
- 692
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Quotient group with $\mathbb{Z}_2$ and $\mathbb{Z}_4$
I recently published another question about quotient groups. In this case, I was practising and trying to find the elements of the group $\mathbb{Z}_2 \times \mathbb{Z}_4 / \langle(1,1)\rangle$. The ...
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Elements of $\mathbb{Z}_6 / \mathbb{Z}_2$
Studying some concepts of group theory and after having read and trying to assimilate the concept of quotient group, I was wondering what structure would have the group $\mathbb{Z}_6 / \mathbb{Z}_2$. ...
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Is it always $|G:gG|=1$ where $G$ is a group and $g \in G$?
Is it always $|G:gG|=1$ where $G$ is a group and $g \in G$?
When $G$ is finite then I use Lagrange’s Theorem,
$$|G:gG|=\frac{|G|}{|gG|}=1.$$
When $G$ is not finite then I don't know what should I do. ...
1
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0
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2-groups with abelianization of type (2,2)
On Olga Taussky's article "A Remark on the class field tower" there is a note on the last page where she states that P. Hall pointed out to her that exists 3 different groups of order $2^k$ ...
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1
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What is meant by the phrase "but the identity in a quotient group is the subgroup" in this Abstract Algebra proof? [duplicate]
I was reading How to Think about Abstract Algebra by Lara Alcock, and encountered the following theorem and proof in the section on Lagrange's Theorem:
Claim: Let $K = \{e, (12)(34), (13)(24), (14)(...
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Element in a coset whose order's prime divisors are those of the order of the coset
This is exercise 3B.6 of Isaacs' "Finite Group Theory"; more specifically, item $(a)$. It goes:
Let $N \lhd G$ and let $g \in G$, where $G$ is a finite group. Suppose $Ng$ has order $m$ in ...
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Meaning of factorization of groups
I am studying group theory from the book "Abstract Algebra" by Dummit and Foote, and in the text the author states the following:
Simple groups, by definition, cannot be "factored"...
1
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1
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98
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Writing a quotient group as a product of cyclic groups
Let $G = \mathbb{Z}_{12} \times \mathbb{Z}_{12}$, and let $a$ be a generator of $\mathbb{Z}_{12}$.
Consider the subgroup $H$ generated by $(a^4, a^6)$. I need to write $G/H$ as a product of cyclic ...
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Isomorphism between isomorphic abelian groups quotiented by isomorphic subgroups [duplicate]
Let $G_1$ and $G_2$ be two abelian groups such that $G_1 \approx G_2$ ($G_1$ is isomorphic to $G_2$). $H_1$ is a subgroup of $G_1$ and $H_2$ is a subgroup of $G_2$ such that $H_1 \approx H_2$. Check ...
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Is every finite simple group a quotient of a braid group?
Question: Is every finite simple group a quotient of a braid group?
Context: The braid group on two strands $ B_2 $ is isomorphic to $ \mathbb{Z} $ and so the infinite family of abelian finite simple ...
- 7,032
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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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If every element of $G/H$ has a square root and every element of $H$ has a square root, then every element of $G$ has a square root.
I have tried to prove this statement but cannot. Does it require the group $G$ to be abelian? This assumption is not stated in the text I am reading, but all I can get is that if we consider that $gH =...
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2
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The abelianizer functor is not faithful
I wanted to understand why the abelianizer functor from the category Grp of groups to the category Ab of abelian groups is not faithful. This amounts to finding a couple of groups $G$ and $H$ and two ...
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Existence of a canonical bijection between $G/H \times H/K$ and $G/K$
Suppose $G$ is a group and $K<H<G$. All of the constructions of a bijection $G/H \times H/K \to G/K$ that I saw go more or less as follows.
Choose a set of representatives ${g_i}$ for $G/H$, ...
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Show group defined via quotient of matrix $A$'s columns has order $|\det A|$.
The problem:
Let $M \leq \mathbb{Z}^n$ be the subgroup generated by the rows of an
$n \times n$ matrix $A$ with entries in $\mathbb{Z}$. Show that $G=\mathbb{Z}^n/M$ is finite if and
only if $\det A \...
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If $G/G’ $ is cyclic, prove by induction that $G/Z(G)$ is cyclic if $G$ is $p$ group. [duplicate]
Let $G$ be a $p$ group. If $G/G’$ is cyclic, prove $G/Z(G)$ is cyclic by induction.
I tried playing with the base cases : if $|G|=p $ or $p^2$, $G $ is abelian, so $G/Z(G)=G/G =\{G\}$ which is ...
- 1,005
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Confusion about the difference between HN/N and H/N in the second isomorphism theorem
Im trying to understand the second isomorphism theorem but I am stuck. So in my textbook the author did the following:
Let G be a group and N<G a normal subgroup and H<G a subgroup. Let $\pi: G\...
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Isomorphic quotient ring implies sum of numerators
I have to show the following proposition:
Let $O$ be a commutative unitary ring that contains $Z$ the ring of integers. $I\le O$ an ideal. Suppose that $\frac{O}{I}\cong\frac{Z}{pZ}$ with p prime ...
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Maximal quotient group of direct product
In GAP small group library, The group $$[32,2]=\langle a,b,c\mid a^4=b^4=c^2=1, ba=abc, [a,c]=[b,c]=1 \rangle.$$
We say $G/N$ is a maximal quotient group if there exists no quotient group $G/K$ such ...
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proof that if G is a solvable group then quotient of G is solvable
I am reading Stewart book on Galois theory, one of the theorem proves that if $G$ is solvable and $N$ is a normal subgroup of $G$, then the quotient of $G$, $G/N$ is solvable. I can't fully understand ...
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$K \leq H_1 \leq H_2 \leq G$, and that $K \trianglelefteq G$ and $H_1 \trianglelefteq H_2$. show that $[H_2 : H_1] = [H_2/K : H_1/K]$.
Suppose that $K \leq H_1 \leq H_2 \leq G$, and that $K \trianglelefteq G$ and $H_1 \trianglelefteq H_2$. Using only Lagrange, show that $[H_2 : H_1] = [H_2/K : H_1/K]$.
(It's not enough to just write ...
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If $[G:K] = m$, then prove that $g^m \in K$ for all $g \in G$.
Let $K$ be a normal subgroup of $G.$ If $[G:K] = m$ then prove that $g^m \in K$ for all $g \in G$.
Solution: If $[G:K] = m$ then $|G/K| = m$. By Lagrange's theorem, the order of an element divides ...
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Solvable-by-finite groups
I am trying to prove this:
Let $G$ be a finite-by-solvable group, i.e. $G$ has a normal subgroup $N$ that is finite with $G/N$ solvable. Prove that $G$ is solvable-by-finite, i.e., $G$ has a solvable ...
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Find the quotient group $G/H$ of group $G = 15\mathbb{Z}_{18}$ by $H = 8\mathbb{Z}_{18}$
I have a question about Quotient groups.
I have a task to find a quotient group $G/H$ of group $G = 15\mathbb{Z}_{18}$ by $H = 8\mathbb{Z}_{18}.$
$G = \{0, 3, 6, 9, 12, 15, 18, 21\}$
I found that $G / ...
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Show that the size of the pre-image $(\psi(x))$ is the same for all $x$. What is this size equal to?
Let $\psi: G \rightarrow H$ be a homomorphism, and let $K=\operatorname{ker}(\psi)$. Define the canonical homomorphism $\varphi: G \rightarrow G/K$ by $\varphi(x) = xK$.
Show that the size of the pre-...
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A quotient group of a $p$-group is a $p$-group [closed]
I'm trying to prove that given a $p$-group $G$ and any normal subgroup $A$ of $G$, the quotient group $G/A$ is also a $p$-group.
If $G$ is finite, then $A$ is finite too and the size of $G/A$ is a ...
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Quotient of an abelian group by torsion subgroup provides a characteristic free abelian subgroup? [closed]
I have a follow up to this question: Rank of the quotient of an Abelian group by its torsion part?.
So my understanding is given a finitely generated abelian group $G$ and its torsion subgroup $T_G$ ...
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Why aren't all surjective continuous homomorphisms of topological groups open?
My apologies if this a naive question.
Let $\phi:G\rightarrow H$ be a continuous surjective homomorphism of topological groups. Denote the kernel of $\phi$ by $N$, then:
$$\begin{align}
f:G\times N&...
- 2,073
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1
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Does it make sense to talk about a factor group by a non-subgroup?
In a problem I've just completed, I show that we have the isomorphism $SL(2,\mathbb{C})/\{-I,I\} \cong \mathcal{M}$ where $SL(2,\mathbb{C})$ is the group of 2x2 complex unimodular matrices, $\{-I,I \}$...
- 1,169
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If the cardinality of the quotient set of a subgroup is $1$ does that mean that the subgroup is equal to the whole group?
Let $(G, \cdot)$ be a group and let $H \leq G$ (i.e. $H$ is a subgroup of $G$) such that $\mid G:H \mid = 1$ (i.e. the index of $H$ in $G$ is $1$). Does this imply that $G=H$?
In the case that $G$ is ...
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Proof that all Quotient Groups are Abelian- where is my error?
I'm having trouble finding the error in my proof:
Theorem: If H is a normal subgroup of G, then the quotient group G/H is abelian.
Proof: $\forall x, y \in G$, $xy(yx)^{-1} = xyy^{-1}x^{-1} = e \in H$....
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1
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Coset of $[18]$ in $\Bbb Z_{55}/11\Bbb Z_{55}?$
Q: Let $G=\mathbb{Z}_{55}$, $H=11\mathbb{Z}_{55}$ and $g=18\pmod{55}.$ Write down the elements of the left coset $gH$ as a comma-separated list of elements of $G$.
My thoughts: First find $G=\mathbb{...
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When do $H_1 \cong H_2$ and $K_1 \cong K_2$ imply $F(H_1,K_1) \cong F(H_2,K_2)$?
Suppose $H_1$ and $K_1$ are subgroups of $G_1$ and $H_2$ and $K_2$ are subgroups of $G_2$ such that $H_1 \cong H_2$ and $K_1 \cong K_2$.
When can we say $H_1 \cap K_1 \cong H_2 \cap K_2$?
If this ...
- 1,993
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How to I take the quotient $GL^+(2,\mathbb{R})/SO(2,\mathbb{R})$
I am using the following representation of the $GL^+(2,\mathbb{R})$ group.
$$
\exp( a+x \sigma_1 +y \sigma_2 + b \sigma_1\sigma_2) = \exp( \begin{bmatrix}a+x & -b +y \\ b+y & a-x\end{bmatrix}...
- 2,455
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Identify the group $(28\mathbb Z + 20\mathbb Z)/20 \mathbb Z$
This is a homework problem for an abstact Algebra course.
Identify the group $(28\mathbb Z + 20\mathbb Z)/20 \mathbb Z$.
I did this just by looking at individual elements of $(28\mathbb Z + 20\mathbb ...
- 2,102
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Order of $xy$ in various quotients of the free product $G_1 * G_2$, where $x \in G_1, y \in G_2$ are nontrivial
Say we have two arbitrary nontrivial groups $G_1$ and $G_2$, and some arbitrary nontrivial elements $x \in G_1, y \in G_2$. Then it is known that the order of $xy$ in the free product $G_1 * G_2$ is ...
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1
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Finitely generated abelian groups $G,H$ with the same finite quotients (up to isomorphism) are isomorphic
Two finitely generated abelian groups $G,H$ have the same finite
quotients (up to isomorphism). Prove they are isomorphic.
According to an earlier result, it seems that I need to use the fact that ...
- 1,649
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1
answer
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For groups, rings, or $R$-modules $A, B, C$ with $C\subseteq B$, does $A/B \cong A/C$ imply that $B=C$?
I think this is true if the module is finite, using Lagrange's Theorem. However, is this the case for infinite modules (or rings, or groups)? This comes from me trying to prove that tensoring is right-...
- 1,042
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1
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inverse elements in quotient ring $\mathbb{Z}/_{\langle 560 \rangle}$
im currently study for exam and I have a problem that I forgot how to solve. I have the quotient ring $\mathbb{Z}/_{\langle 560 \rangle}$.
How do I find how many inverse elementes exists in the ring? ...
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Let $G$ be a subgroup of $E(2)$ and suppose that $T$ is the translation subgroup of $G$. Prove that the point group of $G$ is isomorphic to $G/T$.
$E(n) = \{(A,x):A \in O(n) \text{ and } \mathbf{x} \in \mathbb{R}^n \}$, where $O(n)$ is the group of real orthogonal $n \times n$ matrices
The point group associated to $p \in \mathbb{R}^n$ and a ...
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