Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

2
votes
3answers
42 views

What are the isomorphisms from $\mathbb{Z}$ to itself?

I know there are only two isomorphism from $\mathbb{Z}$ to itself: the identity $Id$ and $-Id$. And I also know we have to use the fact that $\mathbb{Z}$ is cyclic to prove it. However, I have problem ...
3
votes
2answers
51 views

If S is a normal subgroup, identify the quotient group G/S. What are the $\varphi(G)$'s?

The following is an exercise from Artin's Algebra: (Kiefer Sutherland's voice) Let G be the group of upper triangular real matrices $\begin{bmatrix} a & b\\ 0 & d \end{bmatrix}$ with a and ...
0
votes
1answer
23 views

In proving G contains an element of order 15 if contains normal subgroups of orders 3 and 5, respectively, is $HK$ itself cyclic with order 15?

There is an answer here, but it is a "roadmap". group containing normal subgroups of orders $3$ and $5$ contains element of order $15$ There are answers here, but they are "roadmaps" too. If $G$ ...
0
votes
0answers
16 views

Arguing that $(A\cap B)/N = A/N \cap B/N $

Let $N\leq A, B\leq G$ where $N\trianglelefteq G$ so that $G/N$ is a group. With correspondence theorem, I am trying to show that the join and intersection of $A$ and $B$ has a unique correspondence ...
0
votes
1answer
10 views

Circulant Matrices and Ring of Polinomials

I am having problem to proove that the group of circulant Matrices of size nxn, $C_n$ is isomorphic to $\frac{\mathbb{C}[x]}{<x^n-1>}$
0
votes
0answers
44 views

How many (non-isomorphic) abelian groups of order $200$ are there?

I used the fundamental theorem of finite abelian groups. $\mathbb{Z}_{2} \times \mathbb{Z}_{2} \times \mathbb{Z}_{2} \times \mathbb{Z}_{5} \times \mathbb{Z}_{5}$ $\mathbb{Z}_{2} \times \mathbb{Z}_{2}...
0
votes
1answer
23 views

If H is a subgroup of G then show that H and xHx^-1 are isomorphic? [closed]

$xHx^{-1}= \{xHx^{-1} | h \in H \}$ I know how to show that $xHx^{-1}$ is a subgroup but don't know how to show isomorphic.
1
vote
0answers
52 views

why there is no non cyclic group other than $D_{61}$. [duplicate]

I know that there are two non isomorphic groups of order $122$. One is cyclic group of order $122 $ and another is $D_{61}$. I know that all cyclic groups are isomorphic. But I can not understand why ...
6
votes
2answers
218 views

Is it possible to demonstrate two groups are isomorphic without specifying an isomorphism between them?

Usually, when there are two finite groups of small order, we can check if they are isomorphic by trying to impose an isomorphism. But suppose we have two very large finite groups, what are some ...
0
votes
1answer
14 views

Give counterexample to the following claim about invertible homomorphic posets

In general, it is known that all invertible (bijective) group homomorphisms are group isomorphisms. However the same reasoning need not hold true for invertible poset homomorphisms. That is, not all ...
0
votes
1answer
53 views

Establishing isomorphism between the given two groups

Reference (optional) The question essentially is: prove $D_6$ is isomorphic to $S_3 \times \mathbb{Z/2Z}$. To me the linked question doesn't make sense because it seems more like trial and error, by ...
1
vote
1answer
30 views

How to show isomorphism between two given groups? (See details below)

The question (from my exam) quoted verbatim: Show that the additive group of Z[x] is isomorphic to the group of positive rational numbers under multiplication? First of all, is this Z[X] the same as ...
0
votes
0answers
38 views

Show $\phi:H\to HN/N$ is one-to-one.

Can anyone help me with the question? I am stuck on showing this is one-to-one. For subgroup $H$ of $G$ and normal subgroup $N$ in $G$, show the function $\phi:H\to HN/N$ defined by $\phi(h)=hN$ is ...
0
votes
0answers
16 views

Defining a normal subgroup to be maximal.

I just want to clear up something from Question 2. from the following my professor made: For Question 2., if $|G|=p$ and $N$ is maximal, does that mean $|N|\neq p$ since there is no subgroup $H$ of $...
1
vote
2answers
38 views

conceptual question with example: proving groups are isomorphic

The question is to prove $D_8$ and the subgroup of $S_4$ generated by $(1 2)$ and $(1 3)(2 4)$ are isomorphic. I was able to show that the relations for $D_{8}$ follow when we set $b = (1 2)(1 3)(2 4)...
2
votes
3answers
43 views

Show that the groups $SL_2(\mathbb{F}_3)$ and $S_4$ are two nonisomorphic groups of order 24.

I proved that $$SL_2(\mathbb{F}_3)=\left\langle \begin{pmatrix} 1 & 1\\ 0 & 1\end{pmatrix}, \begin{pmatrix} 1 & 0\\ 1 & 1\end{pmatrix}\right\rangle$$ and know that $$S_4=\langle (1\ 2),...
0
votes
4answers
20 views

Finding isomorphism between two automorphism groups.

I am trying to prove "If $G\cong H$, then $Aut(G) \cong Aut(H)$". I've constructed a homomorphism that seemed "natural", but it turns out to be not injective. Is there a methodical way of finding this ...
4
votes
1answer
77 views

Matrix group isomorphic to $\mathbb{Z}$

Define the set $$ G := \left\{ \begin{pmatrix} 1 - 2n & n \\ -4n & 1 + 2n \end{pmatrix} : n \in \mathbb{Z}\right\}. $$ Show that $(G, \cdot)$ is a group using the usual matrix ...
1
vote
1answer
40 views

If 3 is not a divisor of the order of a group, then $\forall g \in G$ $\exists h \in G$ s.t $g=h^3$

Duplicate 1 Duplicate 2 Here, I attempted an alternative proof for a restricted case (when the operation is commutative) Let us define a homomorphism $ \phi:G \rightarrow G $ such that, $\phi (x) =...
1
vote
1answer
50 views

Does the notation $(G/A)/B$ actually make sense?

I'm having a debate with a friend. Let $G$ denote an abelian group, and suppose $A$ and $B$ are subgroups of $G$. The question is whether $(G/A)/B$ makes sense with the standard definitions. My ...
1
vote
2answers
146 views

Additive groups $\mathbb R$ and $\mathbb C$

Are the additive groups $(\mathbb R,+)$ and $(\mathbb C,+)$ isomorphic in Zermelo–Fraenkel set theory with the negation of AC? Added remark: I was told at a lecture that the groups are isomorphic ...
0
votes
0answers
33 views

What are some good concrete examples of Strong Homomorphisms?

I was trying to understand strong homomorphisms conceptually but I was having some difficulties and thought that concrete examples would be useful. I think I understand the definitions but it felt ...
1
vote
1answer
58 views

Is there any isomorphism between the non-zero complex numbers under multiplication and the complex numbers under addition? [duplicate]

It seems like there is no trivial isomorphism between them and both are abelian and non-cyclic, so, it makes it even harder to conclude anything. I need some hints. If I had to make a guess, I'd ...
0
votes
0answers
70 views

Are $\langle \mathbb R,+\rangle $ and $\langle \mathbb R\setminus \{0\},*\rangle $ isomorphic? [duplicate]

I know that $\langle R,+\rangle $ and $\langle\mathbb R^+,*\rangle$ are isomorphic. And Explicit map given by Exponential . I am not able to convince myself for map $\langle \mathbb R\setminus \{0\},*...
2
votes
0answers
60 views

Ambiguity of the notation $\lhd$

If a group $G$ have two subgroups $A,B$ which are isomorphic. Can I say that $A\lhd G$ if $B\lhd G$? Indeed we often do in this way when we don't considerate about the concrete set of groups.
1
vote
0answers
24 views

Isomorphism class of factor groups.

Say $G=Z_4 \oplus U(4), H = \langle (2,3) \rangle, K= \langle (2,1) \rangle$. Determine the isomorphism class of factor groups $G/H$ and $G/K$. I am able to see that $H , K$ are isomorphic but I ...
3
votes
0answers
33 views

Pin group isomorphisms

My goal is to identify the $Pin$ group $$ 1 \to Spin(n) \to Pin^{\pm}(n) \to \mathbb{Z}_2 \to 1 $$ such that $Pin^{\pm}(n)$ are isomorphisms to other more familiar groups. My trick is that to look ...
0
votes
5answers
51 views

A simple question about normal subgroups of finite groups.

Let $G$ be a finite group and let $H_1 \trianglelefteq G$. If $H_2 \leq G$, with $|H_1| = |H_2|$, can we state that $H_2 \trianglelefteq G$? If it is false in general, is the same statement true ...
0
votes
2answers
36 views

Proof about the isomorphism of a non abelian group and the dihedral group

I need to prove the following fact: if $G$ is a non abelian group of order $6$, then $G$ is isomorphic to the dihedral group $D_3$. Here what I have done: Suppose that $G$ is non abelian group of ...
0
votes
1answer
30 views

Any division ring of characteristic $0$ contains a division subring which is isomorphic to $\mathbb{Q}$

In my book, I encountered the following problem: Prove that any division ring of characteristic $0$ contains a division subring which is isomorphic to $\mathbb{Q}.$ I looked this up online and ...
10
votes
3answers
185 views

Affine Transformations isomorphic to Heisenberg group

I want to show that the lie group $G$ of affine transformations of the form $$ \begin{bmatrix} 1 & c & -\frac{c^2}{2} \\ 0 & 1 & -c \\ 0 & 0 & 1 \end{bmatrix} + \begin{bmatrix} ...
0
votes
1answer
71 views

A finite group with exactly $2$ conjugacy classes isomorphic to $\mathbb{Z}_2$

Prove or contradict: A finite group with exactly $2$ conjugacy classes always isomorphic to $\mathbb{Z}_2$. At first I was trying to work with familiar groups to contradict it(permutations, cyclic, ...
3
votes
1answer
37 views

Does $GL_5(\mathbb{R})$ has subgroup of index $2$?

Does $GL_5(\mathbb{R})$ has subgroup of index $2$? My answer - Yes. Let's say that $\tau$ is a function, which is the sign of the determinant of the matrices in $GL_5(\mathbb{R})$ ( easy to see ...
1
vote
0answers
40 views

An infinite group which has only one automorphism [duplicate]

I have known that the only finite group which has only one automorphism is the cyclic group with order less than $2$. But what's the infinite situation? Is there any infinite group which also ...
1
vote
2answers
38 views

Isomorphism between $\mathbb{Z}^2/\ker\phi$ and $\text{im}(\phi)$

The structure $(\mathbb{Z}^2,+)$, where the addition on $\mathbb{Z}^2$ is defined by $(a,b)+(c,d) = (a + c,b+ d)$, forms an additive abelian group. The map $\phi:\mathbb{Z}^2\rightarrow \mathbb{Z}$ ...
4
votes
1answer
65 views

Given $G,L \leq S_4$, if $|G| = |L| = 6$, do we have $L \cong G$

I've been battling at this for a bit, $S_4$ isn't cyclic so that makes things a bit more difficult. I know by Lagrange a subgroup G with order 6 exists. I actually found some examples. Then found by ...
1
vote
0answers
27 views

How could you find the preimage of an isogeny function?

How do you know if an isogeny is surjective or not, and how do you tell how many points on E maps to E'? Does the answer lie in the degree of the isogeny function?
-2
votes
1answer
129 views

Are the isomorphism concepts in Category Theory and Algebra different? [closed]

In Category Theory if there is an $F: A \to B$, and it has an arrrow $G: B \to A$, we can talk about an isomorphism between A and B when $G\circ F=\mathsf{id}_A$ and $F\circ G=\mathsf{id}_B$ In ...
1
vote
3answers
52 views

$\mathbb{Z}_{22}$ surjects onto $\mathbb{Z}_2$

Show that $\mathbb{Z}_{22}$ surjects onto $\mathbb{Z}_2$. I assume that we should use isomorphism theorems here or maybe quotients maps? I tried finding homomorphism for the first isomorphism ...
3
votes
1answer
44 views

Injective homomorphism whose image is contained in the union of two conjugacy classes

Show that there exists injective homomorphism $\tau : \mathbb{Z}_3 \times \mathbb{Z}_3 \times \mathbb{Z}_3 \rightarrow S_{27}$ whose image is contained in the union of two conjugacy classes of $S_{27}...
4
votes
1answer
114 views

Show that $GL_n(\mathbb{F}_{23})$ has subgroup of index $2$

Show that $GL_n(\mathbb{F}_{23})$ has subgroup of index $2$ We know that $|GL_n(\mathbb{F}_{23})|=(23^n-1)(23^n-23) \dots (23^n-23^{n-1})$ and therefore is even. I thought about Cauchy theorem ...
3
votes
2answers
40 views

Does any isomorphism between $\pi_1(X,x_0)$ and $\pi_1(Y,y_0)$ always induce a homeomorphism between $(X,x_0)$ and $(Y,y_0)$?

I know that if $h : (X,x_0) \longrightarrow (Y,y_0)$ is a homeomorphism then that induces an isomorphism $h_{*} : \pi_1(X,x_0) \longrightarrow \pi_1(Y,y_0)$ defined by $$h_{*} ([f]) = [h \circ f].$$ ...
0
votes
1answer
39 views

Isomorphisms of $GF(2)$ [closed]

The additive group of $GF(2)$ is isomorphic to $\mathbb{Z}2/\mathbb{Z}$ under addition with the "carryless" addition taken modulo 2. An appropriate relabelling of the elements ($0 \rightarrow 1$ and ...
3
votes
1answer
54 views

Groups of order $2p$

There are a few question on classification of groups of order $2p$ on MSE but I'd like to receive a feedback on this proof (and have a question about it at the end). Let $G$ be a group of order $2p$. ...
4
votes
3answers
45 views

Show an isomorphism

If $N,M$ are normal subgroups of $G$ then $\frac{NM}{M} \simeq \frac{N}{N\cap M}$ I have been trying to build a function from $N$ to $\frac{NM}{M}$ and looking at is kernel, but i'm.strugling at what $...
1
vote
0answers
38 views

Prove that the cyclic nature of isomorphic groups are preserved

In my textbook, the proof has been done quite in a mainstream fashion. Which goes like the following: Let $\phi:G \rightarrow G'$ be an isomorphism. We are to show that if either one be cyclic, then ...
0
votes
0answers
20 views

On isomorphism of quotient groups of free abelian groups of finite rank

Consider the free abelian group $\mathbb Z^n$, with elements considered as row vectors. For every $A\in M_{r\times n}(\mathbb Z)$ , let $K_A$ be the subgroup of $\mathbb Z^n$ generated by the row ...
-3
votes
3answers
58 views

Prove that $\Bbb{Z}×\Bbb{Z}$ and $\Bbb{Z}×\Bbb{Z}×\Bbb{Z}$ is not isomorphic groups. [closed]

Considering them as rings, I can prove that they are not isomorphic as they have different nos of idempotent elements , but in case of group isomorphism I am unable to prove it.
2
votes
2answers
104 views

Every surjective homomorphism $\alpha : \mathbb{Z}^3 \rightarrow \mathbb{Z}^3$ is isomorphism [closed]

Let $ \alpha : \mathbb{Z}^3 \rightarrow \mathbb{Z}^3$ be a surjective homomorphism. Prove that $\alpha$ is isomorphism. I'm not sure how to approach this... Why does every surjective homomorphism has ...
1
vote
2answers
55 views

On isomorphisms between $G$ and $H\times K$

Here is a proposition from Artin: Usually when one wants to prove that $G\simeq H\times K$, one verifies the conditions of $(d)$. But if $(d)$ holds, then $G$ is isomorphic to $H\times K$ via the ...