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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.

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Groups of order 56 with Sylow 2-subgroup isomorphic $Q_8$

I try to classify non-abelian groups of order $56$ with sylow $2$-subgroup isomorphic to quaterion group $Q_8$. More accurately I want to construct $2$ non-isomorphic such groups. This is an excercise ...
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Permutation groups: $S_4$ and $D_4$.

Question. Determine the subgroup of $S_4$ generated by $\sigma=(1\ 2\ 3\ 4)$ e $\tau = (2\ 4)$. Show that $\left<\sigma, \tau\right> <S_4$ is isomorphic to the group of square symmetries. ...
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Isomorphism between groups of ordre 4 [on hold]

I want to Prove that tow groups of ordre 4 are not isomorph between them
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A group is abelian if and only if $a\mapsto a^{-1}$ is an automorphism [duplicate]

Let $G$ be a Group and show that $G$ is commutative iff $F$ is an automorphism, where $F(a) = a^{-1}$.
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Show that $R^n/Im(\rho)=R^{n-1}\bigoplus R/2R$, where $R$ is an abelian group and $\rho$ is the following function.

Consider the following group homomorphism $\rho$, where $R$ is an abelian group, \begin{align*} \rho:&R\rightarrow R^n\\ \rho(r)=&(2r,2r,\cdots,2r). \end{align*} Show that $R^n/Im(\rho)=R^{n-...
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Show that the quotient group $G/N$ contains a subgroup isomorphic to $H$. [on hold]

Let $G$ be a finite group, $N\mathrel{\lhd}G$ a normal subgroup of $G$, and $H\leq G$ a subgroup of $G$. Suppose that $|H|$ and $|N|$ are relatively prime (i.e., $\gcd(|H|,|N|)=1$). Show that the ...
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If $G$ is a cyclic group of order $d$ with generator $a$, show that $\dfrac{\mathbb{Z}}{d\mathbb{Z}}$ is isomorphic $G$. [duplicate]

We have to $|G|=d$ and $G=\left<a\right>=\{a^n : n \in \mathbb{Z}\}$ by definition of cyclic group. To show what you are asking, I want to define a function, not necessarily this $$\phi : \...
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Group Isomorphism regarding Sylow Subgroups

Suppose I have given two groups say, $G_1,G_2$ such that they have same order.I'm assuming they are non commutative.Then their Syllow subgroups has same order clearly.If I'm given that the number of ...
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$G$ be a group where $G= \mathbb{Z}_6 \oplus \mathbb{Z}_8$ and the normal subgroup $H=\langle(2,4)\rangle$ use orders of elements to determine…

I have the elements of $H$: $\langle(2,4)\rangle={(2,4),(4,0),(0,4),(2,0),(4,4),(0,0)}$ where $|G/H|=8$ possible isomorphic classes: $\mathbb{Z}_8$ ,$\mathbb{Z}_4\oplus \mathbb{Z}_2$, $\mathbb{Z}_2\...
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Finding Homomorphisms from dihedral groups to cyclical groups

Ok so there was another question very similar to this on here however it leaves me a little confused. $\bf{Question}$ Let G = $D_{14}$ the Dihedral group order 14 and A = $c_7$ be the cyclical ...
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1answer
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There are only two groups of order six, up to isomorphism: $\mathbb Z_6$ and $S_3$. [duplicate]

Let $G$ be group with order $6$. Prove that either $G$ and $\Bbb Z_{6}$ are isomorphic binary structure or $G$ and $S_{3}$ are isomorphic binary structure. I know that for isomorphic binary ...
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1answer
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Prove $G/(M\cap N) \cong M/(M\cap N) \times N/(M\cap N)$ where $G=MN$ and $M,N\triangleleft G$

If we consider the map $\phi: M\times N \rightarrow M/(M\cap N) \times N/(M\cap N)$, I was able to show that this is onto and the kernel of the map is $(M\cap N) \times (M\cap N)$ and hence by first ...
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Is $S_4 \times C_2$ isomorphic to $(C_2 \times C_2 \times C_2) \rtimes S_3$

Let $S_n$ denote the symmetric group on $n$ letters and $C_n$ denote the cyclic group of order $n$. Consider $(C_2 \times C_2 \times C_2) \rtimes S_3$ where $S_3$ acts on $(g_1, g_2, g_3) \in C_2 \...
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1answer
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isomorphism of normal subgroup [closed]

I am reading group theory (particularly isomorphism) in the algebra, and stuck on a problem. Hope you guys will help me out: Let $G$ be finite group, and $A$,$B$ be normal subgroups of $G$ such that $...
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1answer
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Proving isomorphism using Cayley

Prove that $\theta$ is a group isomorphism. Let $\theta :\mathbb{Q}^* \rightarrow \operatorname{Aut}(\mathbb{Q})$ $w \mapsto f_w$ and $f_w(x) = wx$ where $w \in \mathbb{Q}^* $ and $x \in \mathbb{Q}$ ...
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1answer
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Isomorphism on spanned lattice

Let $\mathbf x=\left( \begin{matrix}x_1 \\ x_2 \\ \end{matrix} \right)$ and $\mathbf y=\left( \begin{matrix}y_1 \\ y_2 \\ \end{matrix} \right)$ be two linearly independent vectors is $\mathbb R^2$ and ...
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1answer
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Property of isomorphism [closed]

Let $\phi $ be an isomorphism from G onto a group G'.Then For any element prove a and b in G, a and b commute if and only if $\phi(a) $and $\phi (b)$ commute.
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Number of subgroups of $S_6$ isomorphic to $C_3\times C_3$

I have a doubt, I do not know if my method of solving this problem is correct. I have to count how many isomorphic subgroups to $ \mathbb{Z}_3\times \mathbb{Z}_3$ there are in $ S_6 $. being abelian I ...
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Prove $G/H \cong (\mathbb{R}-\{0\})\times(\mathbb{R}-\{0\})$

Let $G$ be the group of all real matrices of the form $\displaystyle\left( \begin{smallmatrix} a & b \\ 0 & c \end{smallmatrix} \right)$ with $ac \neq 0$ under matrix multiplication. Let $H$ ...
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Suppose $H$ and $K$ are normal subgroups of $G$. Prove that $G/H \times G/K$ has a subgroup that is isomorphic to $G / (H∩K)$.

Suppose $H$ and $K$ are normal subgroups of $G$. Prove that $G/H \times G/K$ has a subgroup that is isomorphic to $G / (H∩K)$. Also prove that if $G = HK$, then $G/(H∩K)$ is isomorphic to $G/H \...
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1answer
37 views

If there exists an epimorphism from a group $G_1$ to $G_2$ that is not one-to-one, then can $G_1$ and $G_2$ be isomorphic? [duplicate]

I am trying to show that two groups are isomorphic only if a certain condition holds. I can show that a specific epimorphism between the two groups is one-to-one only if this condition holds, but it ...
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2answers
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Is there a nice way to use the group isomorphism theorems in this proof?

The following theorem is an exercise in the section of a textbook dealing with the group isomorphism theorems. However, I did not use the isomorphism theorems to prove this theorem, so I wonder if ...
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0answers
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How to show $D_3\oplus D_4$ is not isomorphic to $D_{24}$? [duplicate]

How to show $D_3\oplus D_4$ is not isomorphic to $D_{24}$? Here $D_n$ is the dihedral group of order $2n$. I am not sure how to prove this. I am not very good with the dihedreal groups.
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Closed subgroup of $GL({\cal A})$

I'm trying to prove that for any algebra ${\cal A}$, $Aut({\cal A}) = \{a \in End({\cal A})\ |\ a([x, y]) = [a(x), a(y)], \forall x, y \in {\cal A}\}$ is a closed subgroup of $GL({\cal A})$. I can ...
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Isomorphism Question (What is Z9) isomorphic to in U(n)

What is Z9 isomorphic to? I think it is U(10) but I am not completely sure
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Why the empty function is a local isomorphism?

In a study conserning finite model theory, it says that when the two models are purely relational (namely, they contain only predicate symbols and not function as well as constant symbols), the empty ...
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$(\mathbb Z/p \mathbb Z \rtimes \mathbb Z/q \mathbb Z) \times \mathbb Z/q \mathbb Z \cong\mathbb Z/p \mathbb Z \rtimes (\mathbb Z/q \mathbb Z)^2$?

Given: Let $p$ and $q$ be prime numbers such that $q$ divides $p-1$. It is well-know that there is a monomorphism $\varphi: \mathbb Z/q \mathbb Z \to Aut(\mathbb Z/p \mathbb Z)$. Define ...
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Is $H/L$ isomorphic to $K^*\times K^*$ where $K^*=(K_{\ne0},\cdot)$ with $K$ field and $H, L$ certain subgroups of $GL(2,K)$?

Let $K$ be a field, $H=\left\{\begin{bmatrix}a&b\\0&d\end{bmatrix}:a,b,d\in K, ad\ne0\right\}$, $L=\left\{\begin{bmatrix}1&b\\0&1\end{bmatrix}:b\in K\right\}.$ I'm asked to prove that ...
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How to find a homomorphic map in following question?

Let $S1$ and $S2$ be two sets. Suppose that there exists a one-to-one mapping $J$ of $S1$ into $S2$ . Show that there exists an isomorphism of $A(S1)$ into $A(S2)$, where $A(S)$ means the set of ...
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1answer
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Quotient group of $C^*$ by Circle group is isomorphic to $R^+$

$C^*$ is the set of complex numbers except $0$ and $S'$ is the circle group. I have to show the quotient group $C^*/S'$ is isomorphic to $R^+$. Let's define $f:C^*→ R^+$, $z→|z|$. Why is $Kerf$ $S'$? ...
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Identify a group $G$ that has subgroups isomorphic to $\mathbb{Z}_n$ for all positive integer $n$

Identify a group $G$ that has subgroups isomorphic to $\mathbb{Z}_n$ for all positive integer $n$. Source: Gallian Contemporary Algebra, Isomorphism Chapter I am stuck at the very premise of this ...
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1answer
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How definition of isomorphism is defined?

I have observed that isomorphism classes are sensitive to abelianness and cyclic property. Why mathematicians thought those as important to classify? Can anyone explain the motivations for choosing ...
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2answers
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How can we show that as groups $\mathbb Z[x] \times \mathbb Z \cong \mathbb Z[x] \times \mathbb Z \times \mathbb Z$?

How can we show that as groups $\mathbb Z[x] \times \mathbb Z \cong \mathbb Z[x] \times \mathbb Z \times \mathbb Z$? I tried coming up with an isomorphism but I am stuck. One map I tried was $(p(x), ...
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1answer
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Is D4 (the octic group) isomorphic to U(16) (group of units mod 16)?

I know they both have the same cardinality and are not cyclic. As well, I know that both groups have the same number of finite elements. I know of no way to disprove that they're isomorphic, but am ...
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1answer
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How to find set of all automorphism of G in given question without evaluating each bijective map?

Let $G$ be a group of order $4$, $G$ $=$ {$e$,$a$,$b$,$ab$}, $a^2 = b^2 = e , ab = ba$. Determine $A(G)$.Where $A(G)$ is set of all automorphisms of $G$. What i did was - Since we know that for ...
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Using the first isomorphism theorem to prove subgroups

How can I prove that if $H$ is a subgroup of $G$ and $N$ is a normal subgroup of $G$, then $HN$ is a subgroup of $G$ using the first isomorphism theorem? I know how to prove this in general, but my ...
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1answer
36 views

Is the diagram commutative

In the following diagram $f,g$ are isomorphisms and $\pi, \tau$ are canonical projections. Is the diagram then commutative ? $$ \newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-...
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2answers
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How to solve the problem from Topics In Algebra Herstein?

Let $G$ be the dihedral group defined as the set of all formal symbols $x^iy^j$, $i=0,1$, $j=0,1,\ldots,n-1$, where $x^2=e$, $y^n=e$, $xy=y^{-1}x$. Prove The subgroup $N=\{e,y,y^2,\ldots,y^{n-...
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A necessary condition on semidirect products to be isomorphic

Is it true that the semidirect product groups $\mathbb{Z}\ltimes_A \mathbb{Z}^n$ and $\mathbb{Z}\ltimes_B\mathbb{Z}^n$ are isomorphic if and only if $A$ is conjugate in $\operatorname{GL}(n,\mathbb{...
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1answer
14 views

Isomorphism between the dihedral group D2 and the integer group {-1,1}X{-1,1}

I put all of the elements of $D_2$ into matrix form, giving $[\begin{matrix} 1 & 0\\ 0 & 1\end{matrix}]$ , $[\begin{matrix} -1 & 0 \\ 0 & -1 \end{matrix}] $ , $[\begin{matrix} 1 & ...
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1answer
30 views

Finitely presented groups which are neither Hopfian nor cohopfian

Are there any examples of (preferably countable) finitely presented groups which are neither hopfian nor cohopfian? If so, is there a classification of such groups?
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What is wrong with the followig proof that $\mathbb{Z}$ is isomorphic to $\mathbb{Z} \times \mathbb{Z}$ [duplicate]

Since $ G =( \prod_{1}^{\infty} \mathbb{Z} )\times \mathbb{Z} \times \mathbb{Z} \cong (\prod_{1}^{\infty} \mathbb{Z}) \times \mathbb{Z} $, by taking quotients we get $\mathbb{Z \times Z} \...
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1answer
73 views

Show that a subgroup of order $N \in \mathbb{N}$ is isomorphic to $\mathbb{Z}/n\mathbb{Z} × \mathbb{Z}/nn'\mathbb{Z}$. [closed]

The following (from Modular Forms by Diamond Ch.$1$) is mentioned without a proof: If $N$ is the order of $K$ as a subgroup of $\mathbb{Z}/N\mathbb{Z}×\mathbb{Z}/N\mathbb{Z}$ then by the theory of ...
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Let $\phi:\Bbb{Z}_{20} \to \Bbb{Z}_{20}$ be an automorphism and $\phi(5)=5$. What are the possibilities of $\phi(x)$?

Same question here: Possibilities for $\phi(x)$, but I'm striving for a more general solution. Please tell me what I'm doing wrong or if my reasoning is correct. Scratch/solution: $\phi$ is an ...
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0answers
70 views

Two non abelian groups of order pq are isomorphic

So this is a problem from Herstein. My attempt is as follows: Assume p and q are prime and p>q. We know that there exists a unique subgroup of order p ,call it H=(a) and a subgroup of order q,call it ...
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1answer
28 views

Confused about coverings for Lie groups

I'm trying to establish a cover for one lie group (SL(2,C)) to another (SO(3,C)). I can find an isomorphism from the lie algebras sl(2,c) to so(3,c) by taking $ \left[ {\begin{array}{cc} a & ...
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2answers
35 views

A 2-Sylow subgroup of $S_5$ is isomorphic to $D_4$

Here is what I've got so far: Let $H$ be a 2-Sylow subgroup of $S_5$. Since $|S_5|=120=2^3\cdot 3\cdot 5$, thus $|H|=8$. We also know that $D_4$=8. So it seems like I need to find a homomorphism $\phi$...
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1answer
30 views

Consider the set F under the operation of composition of functions ◦.

Let $C = \{z \in \mathbb C \mid |z| = 1\}.$ Let $f_\theta : \mathbb C \to \mathbb C$ be given by $f_\theta (z) = e^{i\theta z}$. Let $F = \{f_\theta | \theta \in \mathbb R\}$. Consider the set $F$ ...
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1answer
21 views

Show linear mapping is isomorphism

The question; show that the linear mapping for which $$ 1 \rightarrow \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, i \rightarrow \begin{bmatrix} i & 0 \\ 0 ...
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Intuitive explanation for: let $I_G$ be the group of > all inner automorphisms of $G$. Then $I_g$ is isomorphic to $G/C_G$

In the book of Fundamental Concepts of Abstract Algebra by G. Ehrlich, at page 106, it is given that Let $G$ be a group with centre $C_G$, and let $I_G$ be the group of all inner automorphisms of ...