Questions tagged [group-isomorphism]

In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. In a sense, the existence of such an isomorphism says that the two groups are "the same."

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28 views

Is there a straightforward group homeomorphism between the 2-adic integers and $[0,1)$?

I think the (set of the) ring of 2-adic integers is represented by: $$\left\{\sum_{i=0}^\infty 2^ix_i:x_i\in\{0,1\}\right\}$$ And I think the set of real numbers in the interval $[0,1)$ is represented ...
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Number of subgroups of $S_4$ isomorphic to $K_4$

I was trying to find the number of subgroups in $S_4$ which are isomorphic to the Klein's four group $K_4$. I know for doing this, I will have to find the subgroups of the type {$e, a, b, ab$} in $S_4$...
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The structure of $\textrm{SL}_2(\mathbb{Z}/4\mathbb{Z})$

In Keith Conard's notes https://kconrad.math.uconn.edu/blurbs/grouptheory/SL(2,Z).pdf Page 8, he has stated a fact that $\textrm{SL}_2(\mathbb{Z}/4\mathbb{Z})\cong A_4\rtimes\mathbb{Z}/4\mathbb{Z}$. ...
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If $P \cap Q = \{1\}$ and $P, Q \triangleleft G$, then $G \cong P \times Q$

I'm trying to prove that given $G$ a group of order $pq$, with $p,q$ primes such that $q \nmid p - 1$ and $p > q$, then $G$ is cyclic. For that, Cauchy's theorem guarantees us that there are $P, Q \...
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Are the groups $\Bbb{Z}_8 \times \Bbb{Z}_{10} \times \Bbb{Z}_{24}$ and $\Bbb{Z}_4 \times \Bbb{Z}_{12} \times \Bbb{Z}_{40}$ isomorphic?

This question is taken from "A first course in Abstract Algebra" by Fraleigh 7th edition, section 11 question 18: Are the groups $\mathbb{Z}_8 \times \mathbb{Z}_{10} \times \mathbb{Z}_{24}$ ...
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Quotient group isomorphic to $\mathbb Q^∗$

It is given that $G=\left\{\begin{pmatrix} a & b\\ 0 & c \end{pmatrix}\text{ with }a,c \in \mathbb Q^∗\text{ and }b \in \mathbb Q\right\}$ is a subgroup of $GL_2(\mathbb Q)$, the group of ...
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$G\cong A\times B$, where $A=\{g\in G\mid g^p=e\}$ and $B=\{g^p\mid g\in G\}$

Let's take an abelian group G, and prime number p. We have the subgroups $A=\{g\in G\mid g^p=e\}$ and $B=\{g^p\mid g\in G\}$. I have to prove that $G\cong A\times B$, with the extra information: G is ...
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$S_n \cong A_n \rtimes \{1,-1\} $

I have seen this question already a few times, but I still do not understand the actual answer. I have to prove the isomorphism between $S_n$ and the semidirect product of $\{1,-1\}$ and $A_n$. I am a ...
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25 views

Show the induced sequence $A\otimes G \to B\otimes G\to C\otimes G \to 0$ is exact, how do we use the cokernel?

For a fixed Abelian group $G$, any map $f:A\to B$ induces a homomorphism $$f\otimes \operatorname{id}\colon A\otimes G \to B\otimes G.$$ Show that this defines a right exact functor, i.e. for any ...
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Aside in Dummit & Foote after the Second Isomorphism Theorem

Let $G$ be a group, let $A \leq G$, and let $B \trianglelefteq G$. Then the Second Isomorphism Theorem says that $AB/B \cong A / A \cap B$. In a remark after the proof (page 98), D&F say that we ...
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Characterize the isomorphisms in $\operatorname{Hom}(G,H)$ < $H^G$ when they exist ( $G , H$ cyclic)

In my lecture notes I have this exercise: Let $G$ and $H$ be cyclic groups. Having defined the operation: $$\varphi \psi: G \rightarrow H: x \mapsto \varphi(x) \psi(x)$$ for which $H^G$ is a group ...
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Proof $\mathbb{C}^* \cong \mathbb{C} / \mathbb{Z}$

I am trying to prove the isomorphism between $\mathbb{C}^*$ and $\mathbb{C} / \mathbb{Z}$. I already established the way to do it: find a surjective homomorphism $f: \mathbb{C} \to \mathbb{C}^*$, ...
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Show that the Group of Real Numbers Excluding $-1$ is Isomorphic to $\mathbb{R}^*$

Let $\langle S, \cdot\rangle$ be the group of all real numbers excluding $-1$ where $a \cdot b = a + b + ab$. Show that $\langle S, \cdot\rangle$ (which will now be shortened to $S$) is isomorphic to $...
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New Criteria of Isomorphism + Abelian Group [duplicate]

The following questions were suggested by my friend, while we were studying fundamental group theory. We had no exact ideas of the way to approach the problems. Questions (1) Let $G$ and $H$ be ...
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Condition of group isomorphism [duplicate]

If $2$ groups $(G,\times)$ and $(G',*)$ have the same number of subgroups of order $k$, for all positive integers $k$ does it mean that they are isomorphic ?
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Natural transformation between group and characters group when restricting to isomorphisms

In Categories for the Working Mathematician 2nd edition, page 17, Mac Lane indicates (I could verify it) that the isomorphism between a finite abelian group $G$ and its characters group $DG = Hom(G, \...
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Solving the Rubik cube with given initial and target states (generalization of standard Rubik cube)

Consider the generalization of the Rubik cube problem, where we are given an initial state $A$ and a final state $B$, and we search for a path between the two. We can easily show that given $A$ and $B$...
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Group operation used in decomposition of Fundamental Theorem of Finite Abelian Groups

As I was taught, any finite abelian group $G$ can be represented up to isomorphism by a direct product of cyclic integer groups of prime power (there is some canonical fuss there, but that's only by ...
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Show $R_{10} $ is isomorphic to $C_4 $

Determine the Residue group $ R_{10} $ whose binary operation is $\times \ mod \ 10$. Show that: $ R_{10} $ has only one element of order 2 $ R_{10} \cong C_4 $ I found that the group $ R_{10} = \...
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Showing that if $\phi:\Bbb{Z}\oplus\Bbb{Z}\to\Bbb{Z}\oplus\Bbb{Z}$ is an epimorphism of abelian groups, then it is an isomorphism.

I am a mathematician working in analysis and my knowledge in algebra is rusty. Is there a direct argument showing that if $\phi:\mathbb{Z}\oplus\mathbb{Z}\to\mathbb{Z}\oplus\mathbb{Z}$ is an ...
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Let $\phi_a$ be the automorphism of $G$ given by $\phi_a(x) = axa^{-1}$. Show that $|\phi_a|$ divides $|a|$.

Assumption: Let $a$ belong to a group $G$ and let $|a|$ be finite. I proceeded as follows: Let $|a|=m.$We have $\phi_a (x)=axa^{-1}$ and so $(\phi_a(x))^2=(axa^{-1})(axa^{-1})=ax(a^{-1}a)xa^{-1}=ax^...
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Cayley Table of $(\mathbb{Z}_5^*, \cdot)$

1) Determine the Cayley Table of $(Z_5^*, \cdot)$ 2) determine which additive group has the exact same table. 3) Further determine an isomorphism between those two groups and prove by means of ...
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Find a splitting field and its Galois group

I'm struggling with this problem and thought maybe I could get some help here :) Consider the polynomial $f(x) = (x^{2} - 2)(x^{2} - 3)(x^{2} - 5)$ over $\mathbb{Q}$ Find the splitting field $E$ of $...
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Isomorphism between $SO_2\tilde{\times}\mathbb{Z}_2$ and $O_2$

This is the exercise 23.10 p. 135 of Groups and symmetry of Armstrong : Let $G$ be an abelian group and write $G \tilde{\times}\mathbb{Z}_2 $ for the semidirect product $G\rtimes_\phi\mathbb{Z}_2$, ...
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isomorphism between a field and non-field rings [closed]

I think that once you have a field and a ring which is not a field, you can conclude that there is no isomorphism between these two. Is it right? if not, is there an example? if true, can someone ...
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Does a deformation retraction of $X$ onto a subspace $A\subset X$ induce an isomorphism $\pi_n(X) \to \pi_n(A)$?

Let's say we have a topological space $X$ and a subspace $A\subset X$. Assume $A$ is a deformation retraction of $X$. Does that imply that the induced homomorphism of the deformation retraction is an ...
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How to prove that some specific group is not isomorphic to any member of any of the 5 families of groups?

For example, let's say we have a group Q8 (Quaternion group). How to prove that this group is not isomorphic to any member of any of the families of groups (Cyclic, Abelian, Dihedral, Symmetric, ...
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76 views

Finding isomorphism between two groups

I'm reading Vern Paulsen's notes. I don't know how can I show $\Bbb R^*/\Bbb R^+$ and $\Bbb Z_2$ are isomorphic because I'm not sure about operation of $\Bbb R^*/\Bbb R^+$. Should I consider it as ...
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Prove that $(\Bbb Z_2 \times \Bbb Z_2\times \Bbb Z_2, +)$ is not isomorphic to $(\Bbb Z_4 \times \Bbb Z_2, +)$

Prove that $(\Bbb Z_2 \times \Bbb Z_2\times \Bbb Z_2, +)$ is not isomorphic to $(\Bbb Z_4 \times \Bbb Z_2, +).$ I believe that both groups have the same cardinality, however, it is not injective as ...
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Grothendieck group “commutes” with direct sum

The Grothendieck completion group of a commutative monoid $M$ is the unique (up to isomorphism) pair $\langle \mathcal{G}(M), i_M\rangle$, where $\mathcal{G}(M)$ is an abelian group and $i_M\colon M\...
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Proving an isomorphism between finitely generated non-trivial subgroup

Okay, I feel like I start every question this way, but I have an idea of the concepts and need some help actually putting it into practice. I'm working on this question from Groups and Symmetry by ...
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For G and H groups, show that (G×{e})◃(G×H), then show that the quotient (G×H)/(G×{e}) is isomorphic to H.

Have no idea where to start :( I think the solution will have to be something like showing that the group (G x {e}) is normal by doing something with left cosets, and isomorphism wouldn't be too hard ...
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Finding $|\!\operatorname{Aut}(L(K_4))|$ using Orbit-Stabiliser Theorem

I know that you can find the size of an automorphism group of a simple graph $G$ by using the Orbit-Stabiliser theorem as follows: let $\DeclareMathOperator{Aut}{Aut}A = \Aut(G)$, and $v$ be a vertex ...
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1answer
42 views

prove that the unit circle is isomorphic to the quotient group $\Bbb R/K$, where $K$ is a normal subgroup of $(\Bbb R, +)$

Prove that the circle group $(S,$x$)$ where $S=\{z∈ \Bbb C | |z|= 1\}$ is isomorphic to the quotient group $\Bbb R/K$ where $K$ is a normal subgroup of $(\Bbb R , +)$ I understand the basic ...
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1answer
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Is this a property of isomorphisms?

In a homework problem I was doing, I was trying to show that $U(8)$ is not isomorphic to $U(10)$. They used that, supposing $f: U(10) \rightarrow U(8)$ was an isomorphism, $|f(3)| = |3| = 4$ and there ...
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1answer
84 views

Show $G\cong \ker(f) \times \mathbb{Z}$ for abelian $G$

First of all, I am aware of the First Isomorphism Theorem but I am not sure how to use it/if it is useful here $G$ is an abelian group and $f:G\rightarrow\mathbb{Z}$ is a surjective group ...
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Is Aut(G×H) isomorphic to Aut(G) × Aut(H) [duplicate]

Please suggest me the proof.Am stuck with it.I saw somewhere that it will be true if (o(G),o(H))=1 but why?
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Showing this quotient group is isomorphic to $\mathbb{Z}_2\times{}\mathbb{Z}_2$

How do I show that $\mathbb{Z}_4\times{}\mathbb{Z}_2/K$ is isomorphic to $\mathbb{Z}_2\times{}\mathbb{Z}_2$, where $K=\langle(2,0)\rangle$. I'm getting confused with the details involved here, I will ...
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1answer
39 views

Do isomorphisms of quotients give isomorphisms of groups? [closed]

I have a group isomorphism $G/\mathbb{Z}\to \mathbb{Z}^{(*n)}$, can I conclude from this fact that $G$ is isomorphic to $\mathbb{Z}^{(*(n+1))}$, the free product of (n+1) copies of $\mathbb{Z}$.
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1answer
55 views

First isomorphism theorem $G=\mathbb{Z}/60\mathbb{Z}=<\overline{1}>$

I have group G and this homomorphism, $\begin {array}{rccl} \phi \colon & G & \longrightarrow & G \\ & \overline{a} &\longmapsto & \overline{a^3}\end{array}$ What is $Im\phi$?...
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Why is it so computationally hard to determine group isomorphism?

Finding an isomorphism requires to show that for 2 groups $G$ and $H$, there exists a bijective map $\phi : G\to H$ such that $$\phi(ab)=\phi(a)\phi(b)$$ For all $a,b \in G$. This is (probably naively)...
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1answer
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Prove that $M_{16} / Z(M_{16}) \cong C_2 \times C_2$

I am trying to answer the following question: Prove that $M_{16} / Z(M_{16}) \cong C_2 \times C_2$. Where $M_{16}=\langle x, y | x^{8}=1, y^{2}=1, yx=x^{5}y\rangle$ is the modular group of order ...
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1answer
44 views

Showing that $\mathbb{Z}[x]/_{(x)}$ is isomorphic to $\mathbb{Z}$

Showing that $\mathbb{Z}[x]/_{(x)}$ is isomorphic to $\mathbb{Z}$, where $(x)$ is an ideal generated by $x$. My attempt: I will try to show that $\psi : \mathbb{Z} \rightarrow \mathbb{Z}[x]/_{(x)}$ ...
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1answer
57 views

Show that $\mathbb{Z}[x]/(x)$ is isomorphic to $\mathbb{Z}$

Let $\mathbb{Z}[x]$ be a ring of polynomials with integer coefficients, $(x)$ be an ideal generated by $x$. Show that $\mathbb{Z}[x]/_{(x)}$ is isomorphic to $\mathbb{Z}$. My attempt: For each ...
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2answers
52 views

Can we find a field, if we have a group?

Lets be $(G,\oplus)$ a group isomorphic to $(\mathbb{R},+)$. Can we find an operator "$\odot$" so that $(G,\odot,\oplus)$ is field? Lets be $(G,\odot)$ a group isomorphic to $(\mathbb{R}^\ast,\cdot)$. ...
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1answer
59 views

The datum of a group is strictly more than the datum of a group up to isomorphism

In a discussion regarding Mochizuki's work on the ABC conjecture which can be found here, Peter Scholze stated that "the datum of a group is strictly more than the datum of a group up to ...
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1answer
35 views

Isometric isomorphism between Hardy Space $h^p(\mathbb{D})$ and $L^p(\mathbb{T})$

I know the question below is a known result but, I would need some help to prove it! Well, I know that in the Poisson integral induces an isometric isomorphism between $L^p(\mathbb{T})$ and the Hardy ...
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1answer
33 views

isomorphisms - subgroup of an specific order (prove)

I've got that G is isomorphic to C_6 x C_2 , and I have to prove that G has got only one subgroup (I'm going to call H to that subgroup) of order 3. I've done the following: I've taken an element ...
2
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0answers
57 views

Are the two groups $\mathbb R$ and $\mathbb C$ isomorphic? [duplicate]

The two groups $\mathbb R$ and $\mathbb C$ are isomorphic. Solution: We know that $\mathbb {R(Q)}$ and $\mathbb {R^2(Q)}$ are isomorphic as vector spaces. So,there is a vector space isomorphism ...
3
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1answer
71 views

Prove that $G(\mathbb{Q}(\zeta)/\mathbb{Q}) $ and $\mathbb{Z}_n^\times$ are Isomorphic

Prove that $G(\mathbb{Q}(\zeta)/\mathbb{Q}) \rightarrow \mathbb{Z}_n^\times$ is an isomorphism. $\mathbb{Q}(\zeta)$ is the cyclotomic extension of $\mathbb{Q}$ ($\zeta$ is a root of $x^n - 1$). $\...

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