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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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Show that if N, K are normal subgroups of a group G, and N contains K then we have: $ G / N \cong (G/K) / (N /K) $ [duplicate]

Show that if $N, K$ are normal subgroups of a group $G$, and $N$ contains $K$ then we have: $$ G / N \cong (G/K) / (N /K) $$ Intuitively it looks correct, would like to know how I can approach this.
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Is it necessary that $f: G_1 \to G_2$ is an isomorphism, for $f:H\to H$ an automorphism with $G_1, G_2\le H$ of the same cardinality?

This might be a very trivial question, and I have explained what I think about it below. Let's say I have an automorphism $f : H \to H$. Now, let's say I take two subgroups $G_1$ and $G_2$ of $H$. Is ...
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A proof about Automorphism in congruence class

Suppose $gcd(m,n)=1$, and let $F :Z_n→Z_n$ be defined by $F([a])=m[a]$. Prove that $F$ is an automorphism of the additive group $Z_n$. I find it is diffcult to prove $F$ is injective and surjective. ...
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Prove that the central product of $Z_4$ and $D_8$ is isomorphic to $Z_4$ and $Q_8$.

Let $Y_1=\{(1,1),(x^2,r^2)\},\;Y_2=\{(1,1),(x^2,-1)\}$, then prove $Z_4\times D_8/Y_1\cong Z_4\times Q_8/Y_2$. Since both of them have order $16$, it's not hard to list all of their elements, but is ...
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Direct sum of $n$ (infinite) cyclic groups isomorphic to direct sum of $n$ copies of $\mathbb{Z}$?

I'm currently selfstudying some algebra and i am currently covering the various equivalent definitions of free abelian groups. However, in order to understand why these definitions are indeed ...
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Isomorphism factor by factor

Consider groups are finite. Let $G_1 = A \rtimes_{\phi_1} B_1$ and $G_2 = A_2 \rtimes_{\phi_2} B_2$. Note that $A_1,A_2,B_1,B_2$ are cyclic groups. It is also known that $A_1 \cong A_2$ and $B_1 \...
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Given $H \leq G$, prove isomorphism of $H$ and $gHg^{-1}$

All I am given is that $H \leq G$ and I have to prove that $H \cong gHg^{-1}$. I first verified that $gHg^{-1} \leq G$. Then, I tried the map $f(h) = ghg^{-1}$ and it indeed turned out to be an ...
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2answers
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Is $D_n$ isomorphic to $C_n \times C_2$?

I am doing a proof for an exercise where if the dihedral group $D_{n}$ $\cong$ $C_{n} \cdot C_{2}$ where $\cong$ means isomorphic and $\cdot$ is the direct product, I would be able to finish it, but I ...
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Let $A_1$, $A_2$ be groups and $\phi:A_1\to A_2$ surjective with $G_1\unlhd A_1$ and $\phi(G_1)=G_2$. Is $A_1/G_1\cong A_2/G_2$? [duplicate]

Let $A_1$ and $A_2$ be a group and $\phi: A_1 \to A_2$ be a surjective group homomorphism. Also, $G_1 \unlhd A_1$ and $\phi(G_1) = G_2$. I have to prove/disprove this statement: Is $A_1/G_1 \cong A_2/...
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Is there any injective homomorphism (i.e. monomorphism) from a non-cyclic group of order $4$ to $\mathbb{Z}_8$?

The only such possible group is $V$ (up to isomorphism). If $\phi$ be such an into homomorphism, then $o(\phi(V))=4$ and $\phi(V)$ being a subgroup of $\mathbb{Z}_8$, it must be cyclic with a ...
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Isomorphism problem for the center of modular group algebras

Let $p$ be a prime number, $G,H$ a finite p-Group and $K$ a finite field with $\operatorname{char}(K)=p$. It is well-known that the group $1+\operatorname{rad}(KG)$ is a p-group containing $G$. My ...
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How do I find the kernel of a given group homomorphism?

I am given a group $H$ and two normal subgroups $A$ and $B$ and $H/A$ and $H/B$ are two quotient groups. A homomorphism is defined as follows: $\phi: H \to H/A \times H/B; \phi(h) = (hA,hB)$. How do ...
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1answer
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Prove that subgroup of all elements of finite order in group $(\mathbb{C} \setminus \{0\}, \cdot)$ is isomorphic to $\mathbb{Q}/\mathbb{Z}$

Let $H$ be the subgroup of all elements of finite order in group $\left(\mathbb{C} \setminus \{0\}, \cdot \right)$. Prove that $H$ is isomorphic to $\mathbb{Q}/\mathbb{Z}$, where $\mathbb{Q}$ and $\...
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Mapping from $\mathbb Z/2\mathbb Z \times \mathbb Z/3\mathbb Z \times \dots\times \mathbb Z/p_n\mathbb Z$ to $\mathbb Z/p_n\#\mathbb Z$.

I know $\mathbb Z/2\mathbb Z \times \mathbb Z/3\mathbb Z \times \dots\times \mathbb Z/p_n\mathbb Z$ is isomorphic $\mathbb Z/p_n\#\mathbb Z$ (where $p_n\#$ is the primorial of primes up to $p_n$) by ...
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If $(m, n) = 1$, then $\dfrac{\mathbb Z \times \mathbb Z}{\langle (m,n)\rangle}\cong\mathbb Z$

How to show that If $(m, n) = 1$, then $\dfrac{\mathbb Z \times \mathbb Z}{\langle (m,n)\rangle}\cong\mathbb Z$. (Note: We consider this in group theory.) I know that $(m, n) = 1$ means that ...
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Proving the given $\mathbb R^3/H$ $\cong$ $\mathbb R^2$ where $H$ = {$(y,0,0)|y \in \mathbb R$}

So I am given a group $\mathbb R^3$ and a group $H$ = {$(y,0,0)|y \in \mathbb R$}. I have to prove that that $\mathbb R^3/H$ $\cong$ $\mathbb R^2$. I am not sure how to even begin. My difficulty is ...
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1answer
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Find subgroup of $GL(n,\mathbb{Z}_{19})$ with index 3

Find subgroup of $GL(n,\mathbb{Z}_{19})$ with index 3 My try: I thought about looking at $\det:GL(n,\mathbb{Z}_{19}) \to\mathbb{Z}_{19}^*$ and finding a subgroup of index 3 of $\mathbb{Z}_{19}^*$. ...
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1answer
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dφ is bijective but φ is not a lie group isomorphism

suppose G and H connected Lie groups. Is there $\phi: G \to H$ a morphism of Lie groups such that $d\phi$ is bijective but $\phi$ is not an isomorphism of Lie groups? I know that $d\phi$ surjective ...
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4answers
60 views

What's an isomorphism between $Z_{p^2}^*$ and $Z_p\times Z_{p-1}$

Is it true that $Z_{p^2}^* \simeq Z_p\times Z_{p-1}$? One can verify that $|Z_{p^2}^*|=p(p-1)$. Can you give an isomorphism?
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Step in proof of “Does $G\times K\cong H\times K$ imply $G\cong H$?”

I would like to know the idea behind the induction proof in the this question. I added a comment to the answer, but the author has not been active for a year.
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1answer
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Proving the image of a group homomorphism is a subgroup of its codomain

Can anyone help me prove this first step: given that $\varphi : G \to H$ is a group homomorphism, I seek to prove that $\varphi(G)$ is a subgroup of $H$. I'm working on the First Isomorphism theorem ...
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2answers
135 views

Should the isomorphism theorems be seen as an “interface” between algebra and category theory?

My first instinct when I thought about algebra in category theory, was to try to "generalize the isomorphism theorems in category theory". So I tried to prove the generalization of "the image of a ...
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1answer
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Prove $(G \times \mathbb{Z}_2)/K \cong G$ for $K = \{(e, 0), (g, 1)\}$ and $o(g) = 2$ with $g \in Z(G)$.

I'm a bit stuck on this question, any help will be appreciated, here's the question: Let $G$ be a group with an element $g \in G$ such that $o(g) = 2$ and assume $g \in Z(G)$. Let $K$ be a subset ...
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Homomorphism $\phi: G \rightarrow H$, $\phi$ surjective, $\exists a \in H : |a| = 5$, show $\exists x \in G : |x| = 5$

Homomorphism $\phi: G \rightarrow H$, $\phi$ surjective, $\exists a \in H : |a| = 5$, show $\exists x \in G : |x| = 5$, where $|G|$ is finite. I'm not sure if this proof is correct, but here's what I ...
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1answer
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Relation between the order of the elements of 2 groups and isomorphism

Let be $G$ and $H$ two equinumerous groups of order $n$. We label the elements in such a way that $$ 1< |g_1|\le|g_2|\le \cdots \le |g_n| $$ $$ 1< |h_1|\le|h_2|\le \cdots \le |h_n| $$ If $(|g_1|...
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Number theory in Cryptography: Can I say two following subgroups are isomorphic?

I am a cryptographer. for designing an encryption system, I need some number theory/algebraic conditions to be satisfied. So I know what I need, but I dont know if they are really satisfied in the ...
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2answers
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Splitting field/subfield is isomorphic

(1)Let $h$= $\mathbb{Q}$[t]/($t^2-2$). Show that there exists only one subfield of $\mathbb{R}$ isomorphic to $h$. (2)Let $h$= $\mathbb{Q}$[t]/($t^3-2$). Show that there exists three(3) subfields of ...
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If $H$ is a subgroup of $G$ and $a \in G$. Show that $H$ and $a^{-1}Ha$ are isomorphic.

I feel like I did something wrong, because I never actually used $a \in G$. Proof: Define $\alpha: H \rightarrow a^{-1}Ha$ by $\alpha(h)=a^{-1}ha$ Let $x,y \in H$ then, $\alpha(xy) = a^{-1}xya\\ =...
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2answers
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Proving that something is an isomorphism between group generators.

So I'm having trouble understanding isomorphisms between generators. In this problem, I had to show that a non-abelian group of order $6$ is isomorphic to $S_3$. Now I already showed that there were ...
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1answer
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Is $\prod_{i=1}^\infty \Bbb Z\times(\mathbb{Z}\times\Bbb Z)\cong \prod_{i=1}^\infty\Bbb Z\times(\Bbb Z)$ ? Yes/no

i found this question Show by example that this need not be true if we do not assume that the groups are finitely generated but i have doubts in my minds Let $G, H,$ and $K$ be finitely generated ...
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2answers
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Similarities between the idea of Group Homomorphisms and Linear Transformations

I am not sure if this question has been asked before, but my search did not return me any answers. While reading several online notes that attempt to give an intuitive understanding of group ...
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1answer
56 views

Question about the proof of Second Isomorphism Theorem

The Second Isomorphism Theorem: Let $H$ be a subgroup of a group $G$ and $N$ a normal subgroup of $G$. Then $$H/(H\cap N)\cong(HN)/N$$ There is the proof of Abstract Algebra Thomas by W. Judson: ...
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2answers
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Prove if the the groups $A_4$ and $S_3 \times \Bbb Z_2$ are or not are isomorphic

I'm trying to check if the groups $A_4$ and $S_3 \times \Bbb Z_2$ are or not isomorphic. How can I check if they are? I'm trying to understand how can I generally prove an isomorphism with this kind ...
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1answer
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Proof explanation of $P(p^e) \cong \mathbb{Z}/(p-1)\mathbb{Z} \times \mathbb{Z}/(p^{e-1})\mathbb{Z}$.

The following is from Classical Theory of Algebraic Numbers by Paulo Ribenboim : $P(p^e)$ is the set of all nonzero residue classes a modulo m, where gcd(a, m) = 1. My question underlined and in ...
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1answer
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Isomorphism between $GL_2(\mathbb{F}_2)$ and $S_3$ [duplicate]

The question is to show that $GL_2(\mathbb{F}_2)$ and $S_3$ are isomorphic where $S_3$ is the symmetric group of $\{1, 2, 3\}$ and the group operation is composition. I have listed all the elements ...
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0answers
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Internal Semidirect with Factors Isomorphic to “Outside” Groups

Here is a conjecture of mine: If $G$ is the internal semidirect product of $N \unlhd G$ and $Q \le G$, and $\phi_1 : N' \to N$ and $\phi_2 : Q' \to Q$ are isomorphisms, then there is some $\theta :...
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1answer
38 views

Index of a subgroup is preserved under group isomorphism

Let $\varphi : G\to G'$ be a group isomorphism and let $N\lt G$. prove: if $[G:N]\lt \infty$ then $[G:N]=[G':\varphi(N)]$ It is known that $\varphi(N)$ is indeed a subgroup of $G'$, but how can ...
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0answers
28 views

Give three non-isomorphic groups of order 24, and explain why they are not isomorphic. [duplicate]

I am not sure how to do this question, any help would be very much appreciated!
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2answers
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Let $G,H$ be groups and $\varphi:G \times H\to G$ and $H'=\ker(\varphi)$. Show that $(G\times H)/H'\cong G$

This is Exercise 10 from Section 7: Groups and Homomorphisms, Chapter 1: Foundation, textbook Analysis I by Herbert Amann and Joachim Escher. Let $(G,\odot)$ and $(H,\circledast)$ be groups, and ...
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2answers
51 views

Let $H=\{\alpha \in S_7 :\alpha(3)=3\}$ and $K=\{\alpha \in S_7:\alpha(5)=5\}$. Prove that $H\cong K$.

Question: Let $H=\{\alpha \in S_7 :\alpha(3)=3\}$ and $K=\{\alpha \in S_7:\alpha(5)=5\}$. Prove that $H\cong K$ I'm not really able to do much here, I know I have to find some isomorphism but I'm ...
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1answer
26 views

Left representation is isomorphic to group

Let $G$ be a group and $G_L$ be its left representation, that is $G_L = \{g_L ; g_L(x)=gx\}$. Show that $G$ is isomorphic to $G_L$. Solution To show that $G$ is isomorphic to $G_L$ we need an ...
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3answers
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Group-subsets of monoids with different identities

A subgroup must have the same identity as its containing group, but this fact requires inverses. I'm interested in subsets of monoids, which are groups in their own right, but vary greatly from the ...
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1answer
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Prove that $G_2=\{B_1,B_2,…,B_p\}$ is a multiplicative group isomorphic with $G_1=\{A_1,A_2,…,A_p\}$

Let $m,n,p\geq3$ be natural numbers and $G_1=\{A_1,A_2,...,A_p\}$ a multiplicative group of order $p$ with elements from $M_2(\mathbb{Z})$. For every $A_i$ from $G$ we attach $B_i$ such that if $A=(a_{...
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1answer
27 views

Non-trivial isomorphism between the dihedral group to itself.

I want to find a non-trivial isomorphism between the dihedral group $D_n$ and itself. Non-trivial means that the isomorphism won't be the identity. I looked at the group $D_n$ as the set of the ...
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2answers
59 views

What is a local group isomorphism?

What does it mean for 2 groups to be locally isomorphic? E.g. $SO(4)$ is locally isomorphic to $ SO(3)\times SO(3)$ -why not globally?
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0answers
34 views

Find all the natural numbers such that $Z_n$ is isomorphic to the dihedral group $D_n$

The task is to Find all the natural numbers such that $Z_n$ is isomorphic to the dihedral group $D_n$. I am pretty sure that for $n>2$ there is no such isomorphism because $Z_n$ has $n$ elements, ...
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1answer
72 views

In a cyclic group with subgroup $H$ and $N$, list the elements in $H + N$ and $H \cap N$ [closed]

In the Group $G= \mathbb{Z}/24\mathbb{Z}$, let $H = \langle4 + 24\mathbb{Z}\rangle$ and $N = \langle 6 + 24\mathbb{Z}\rangle$. List the elements in $H + N$ and $H \cap N$ I am aware that $G$ is the ...
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1answer
68 views

What is an outer automorphism? [closed]

I know that an automorphism of a group is just an isomorphism of that group with itself. I know that an inner automorphism is an isomorphism/automorphism of the form $\phi_g$ where $\phi_g(x) = gxg^{-...
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3answers
41 views

Finding an isomorphic subring of matrices

I'm struggling a fair amount with this exercise: Find a subring of $M(2,\mathbb{Q})$ which is isomorphic to a) $\mathbb{Q}$ x $ \mathbb{Q}$ b) $\mathbb{Q}$ c) $\mathbb{Q}[x]$/$x^2$ Now I know a ...
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3answers
60 views

Is $f(x)=k \log x$ the only solution to the equation $\sum_{i=1}^{n}f(x_{i})=f(\prod_{i=1}^{n}x_{i})$?

For $x_{i}\in \mathbb{R}_{+}$ Is $f(x)=C\log x$ $(C\in \mathbb{R})$ the only solution to the equation: $\sum_{i=1}^{n}f(x_{i})=f(\prod_{i=1}^{n}x_{i})$ I have tried to solve this problem in the ...