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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. In a sense, the existence of such an isomorphism says that the two groups are "the same."

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The Uniqueness of the Logarithm as a Group Isomorphism between the Positive Reals and Reals

My Group Theory textbook asks of me that I prove the following statement: If $\varphi: \mathbb{R}^{+} \rightarrow \mathbb{R}$ be any isomorphism between the groups $(\mathbb{R}^{+}, \cdot)$, $(\...
Barbatulka's user avatar
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Show That For Every Finite Group $G$ and Field $𝔽$ - $G$ is Isomorphic to a Sub-Group of $GL(V)$ for Some finite-dimensional Vector Space $V$ over F

I need to show that if a group $G$ is finite, then for every field $𝔽$ there's a finite-dimensional vector space $V$ over it such that $G$ is isomorphic to a sub-group of $GL(V)$. I figured the ...
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1 answer
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Describe all non-isomorphic groups of order $57$

Describe all non-isomorphic groups of order $57$, such that for each of them you write down its generators and the connections between them. Attempt: 57 is the product of two primes, specifically $57 =...
lolip123's user avatar
12 votes
1 answer
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Does there exist a group $G$ such that $\operatorname{Aut}(G)\cong D_5$, where $D_5$ denotes the dihedral group of order 10?

I came across this problem stated in the title having no clue what to do, and got stuck even in the finite case. Here's my attempt: I first proved that the group $G$, if it exists, cannot be finite ...
Cyankite's user avatar
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3 votes
1 answer
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If $G/Z(G)$ is isomorphic to a subgroup of $\mathbb Q$ then $G$ is abelian.

I want to show the following two statements: a) Let $G$ be a group such that $G/Z(G)$ is isomorphic to a subgroup of $\mathbb Q$ then $G$ is abelian. For part a) I have the following facts "If $G/...
Fuat Ray's user avatar
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4 votes
1 answer
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Isomorphism between linear functionals $\mathbb{L}(\mathbb{R}^2)$ and $\mathbb{R}^2$.

Problem: Show that if $\mathbb{L}(\mathbb{R}^2)$, the space of the linear functionals in $\mathbb{R}^2$, with the matrix norm $\|\cdot\|_p$ ($p > 1$) is isomorphic with the $\mathbb{R}^2$ space ...
Wellington Silva's user avatar
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Extending bijection on generators to isomorphism of groups

Given two group presentations $G=\langle X \vert R\rangle$ and $G=\langle Y \vert S\rangle$, let $f:X\to Y$ be a set function only on the generators such that $f(x_1)\ldots f(x_n)=1$ for $x_1\ldots ...
Mithrandir's user avatar
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3 answers
71 views

Is $(\mathbb{Q},0,+,<)$ isomorphic to $(\mathbb{Q}^{+},1,\cdot,<)$? ("Logic in Mathematics and Set Theory" by Kazuyuki Tanaka and Toshio Suzuki)

I am reading "Logic in Mathematics and Set Theory" by Kazuyuki Tanaka and Toshio Suzuki. Problem 1.17 Is $(\mathbb{Q},0,+,<)$ isomorphic to $(\mathbb{Q}^{+},1,\cdot,<)$? My attempt: ...
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Is the direct product of finitely-generated groups cancellative? [duplicate]

The direct product is cancellative for finite groups, so I wanted to know if this result holds for finitely-generated groups as well. The proof linked clearly doesn't apply there, but I have been ...
Zoe Allen's user avatar
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Prove that $S_4/K \cong S_3$ using the fundamental theorem on homomorphism.

Let $A$ be the set formed by the elements of Klein group but identity, A= { (1,2)(3,4);(1,3)(2,4);(1,4)(2,3)}. Consider the set $Big(A)$ of bijection from A to itseld. With the operation of ...
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3 votes
0 answers
62 views

Natural map of automorphism groups

Question: Write $\mathbb{Q}(\zeta_{\infty}) = \mathbb{Q}(E)$, where $E$ is the group of roots of unity in $\mathbb{Q}^{*}$. Prove that $\mathbb{Q} \subset \mathbb{Q}(\zeta_{\infty})$ is Galois, and ...
ByteBlitzer's user avatar
3 votes
2 answers
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Use of correspondence theorem for groups to prove that $o(N) = 2$

Let $G$ be a group and $H \triangleleft G$ simple such that $[G : H] = 2$. I have to prove that if $N \neq \{1\}$, $N \triangleleft G$ and $N \cap H = \{1\}$ then $o(N) = 2$. I know by third ...
Cyclotomic Manolo's user avatar
0 votes
1 answer
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Screwing up basic short exact sequence $0\to\mathbb Z\leftrightarrow\mathbb Z\to0$

Represent an isomorphism by $\leftrightarrow$. HAVE exact sequence. $$ 0 \rightarrow \mathbb Z \leftrightarrow \mathbb Z \rightarrow 0 $$ Then $$ \text{img} \left( 0 \rightarrow \mathbb Z \right) = 0 =...
Nate's user avatar
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8 votes
1 answer
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Isomorphism of topological groups

Question: Let $\mathbb{Q}(\sqrt{\mathbb{Q}})$ be the subfield of $\overline{\mathbb{Q}}$ generated by $\{\sqrt{x} : x \in \mathbb{Q}\}$. Prove that $\mathbb{Q} \subset \mathbb{Q}(\sqrt{\mathbb{Q}})$ ...
ByteBlitzer's user avatar
6 votes
0 answers
100 views

Commutativity of the wreath product

Let $G$ be a subgroup of the symmetric group $\mathfrak{S}_n$ and $H$ be a subgroup of $\mathfrak{S}_m$. Recall that the wreath product $G \wr H$ is the semi-direct product $G^m \rtimes H$, where $H$ ...
eti902's user avatar
  • 766
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1 answer
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Computing whether two finite groups are isomorphic (in C++) [closed]

I need to algorithmically compute whether two given finite groups are isomorphic. Usually I only have generators of these groups. The groups can get quite large as I'm working with subgroups of $S_{32}...
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For some finite group $G$ assume $X,Y \subset G, X \cap Y = \emptyset$. Compute subsets of $X$ and $Y$ that generate isomorphic subgroups of $G$.

For some finite group $G$ assume $X,Y \subset G, X \cap Y = \emptyset$. I need to compute subsets $X' \subset X$ and $Y' \subset Y$ that generate isomorphic subgroups of $G$: $\langle X' \rangle \leq ...
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0 votes
1 answer
33 views

Let $A=\langle\{a_1,\dots,a_k\}\rangle,B=\langle\{b_1,\dots,b_k\}\rangle(ord(a_i)=ord(b_i), i=1,\dots,k)$. If $|A| = |B|$, are $A$ and $B$ isomorphic?

Let $A=\langle\{a_1,\dots,a_k\}\rangle,B=\langle\{b_1,\dots,b_k\}\rangle$ where $a_i$ and $b_i$ are elements of some group and $ord(a_i)=ord(b_i), i=1,\dots,k$. In general, if $|A| = |B|$, are the ...
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For $M,C\subset S_n,|M|=|C|$ find subsets of $M$ and $C$ that generate isomorphic subgroups of $S_n$ and the isomorphism maps these subsets together

I have sets $M,C \subset S_n$, s.t. $|M| = |C| \gg 1$. Given $a \in S_n$ I can determine whether $a \in M$ but I have no way to determine whether $a \in C$, however, if we assume C is numbered for ...
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  • 729
1 vote
0 answers
55 views

Is a Rational Map Possible Between Elliptic Curves of Orders 19 and 18?

I'm exploring the relationship between elliptic curves and their orders. I have two elliptic curves: E1: An elliptic curve with order 19. E2: An elliptic curve with order 18. Since the order of E1 is ...
Salad's user avatar
  • 11
2 votes
1 answer
80 views

A question on an isomorphism between ${\rm PSL}_2(9)$ and $A_6$ [closed]

I found a nice argument proving that ${\rm PSL}_2(9)\cong A_6$ on page 52 of The finite simple groups by Prof. R.A. Wilson. Let $f:A_6\to S_{{\rm PL}(9)}$ with $(123)^f= z\mapsto z+1$, $(456)^f= z\...
Probability enthusiast's user avatar
1 vote
2 answers
49 views

Question over decomposition of $\mathbb{Z}_{mn}$

If $m<n\in\mathbb{N}$ and $(m,n)=1$, then there is a natural isomorphism $h: \mathbb{Z}_{mn}\to \mathbb{Z}_m \times \mathbb{Z}_n$. But I'm a little confused about what happens when multiplying $m$ ...
user760's user avatar
  • 1,670
0 votes
1 answer
50 views

Every abelian quotient group is a quotient group of $G/G'$ [duplicate]

A group theory book (not in English) I'm reading states the following: $G/G'$ is the largest abelian quotient group. In fact, every other abelian quotient group is also a quotient group of $G/G'$, ...
M_N1's user avatar
  • 149
0 votes
0 answers
16 views

Determining if the three-equation model is isomorphic to the three-point plane model

I'm self studying off of the textbook "Axiomatic Geometry" by John Lee. During the text, the author mentions of the following models of incidence geometry: Three-point plane model (1) A &...
Aryaan's user avatar
  • 281
0 votes
1 answer
24 views

proving that $A_n/\operatorname{ker} \beta \cong \operatorname{ker} \alpha_{n - 1}.$

Prove that in any exact sequence $$\dots \xrightarrow{\alpha_{n+3}}A_{n+2} \xrightarrow{\alpha_{n+2}} A_{n+1} \xrightarrow{\alpha_{n+1}} A_{n} \xrightarrow{\alpha_{n}} A_{n -1}\xrightarrow{\alpha_{n-...
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2 votes
1 answer
146 views

Isomorphism theorem misunderstanding

One of the isomorphism theorems states $(HN) \ / \ N \cong H \ / \ (H\cap N)$. I am confused about the first part $(HN) \ / \ N$, and whether it is equivalent to $H \ / \ N$. Every element in $HN$ is $...
nezam jazayeri's user avatar
0 votes
0 answers
37 views

Consider the function $\phi(x)=\ln x$ from $\mathbb R^+$ (under mult) to the group $\mathbb R$ (under add) . Prove that $\phi$ is an isomorphism. [duplicate]

Okay so I've proved all of the parts needed for an isomorphism except to show that $\phi(x)$ is onto. How do I go about doing that? I know that you start by assuming that $y \in \mathbb{R^+}$ and that ...
Whole Milk Slammer's user avatar
3 votes
2 answers
86 views

Are these two groups $\mathbb{Q}[x]$ and $\mathbb{Q}$ under addition isomorphic or not?

I am a beginner at ring theory, and after studying the polynomial rings, I have known about the polynomial ring $\mathbb{Z}[x], \mathbb{Q}[x], \mathbb{R}[x]$ which forms a group under addition. After ...
RITAM SADHUKHAN's user avatar
3 votes
2 answers
90 views

Subgroup of braid group $B_3$ isomorphic to itself

Consider the braid group $$B_3=\langle x,y:xyx=yxy\rangle.$$ It has a proper subgroup $N$, defined as follows: $g$ is in $N$ if and only if the sum of all exponents in $$g=\prod u_i^{\varepsilon_i},\ ...
atzlt's user avatar
  • 562
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0 answers
39 views

For groups $F, G, H$, if $G \not\approx H$, does that mean $F \oplus G \not\approx F \oplus H$? [duplicate]

For context, we were asked to prove in my Abstract Algebra class that $$\mathbb{Z}_3 \oplus \mathbb{Z}_9 \not\approx \mathbb{Z}_3 \oplus \mathbb{Z}_3 \oplus \mathbb{Z}_3, \tag{1}$$ where $\mathbb{Z}_n$...
Mailbox's user avatar
  • 927
5 votes
3 answers
1k views

Is this a valid "easy" proof that two free groups are isomorphic if and only if their rank is the same?

I have read on different sources that it is not possible to give a simple proof that "two free groups are isomorphic if and only if they have the same rank" using only what "a student ...
agv-code's user avatar
4 votes
1 answer
120 views

A group isomorphic to $\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_3$

I am taking a first course on group theory. I understand why $\mathbb{Z}_2\times\mathbb{Z}_3 \cong \mathbb{Z}_6$. How can I use this fact to show that $\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_3 ...
Fuzzy's user avatar
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1 vote
1 answer
62 views

Show that $f(x, y)=(-1)^y x$ is an isomorphism from $\mathbb{R}^+\times \mathbb{Z}_2$ to $\mathbb{R}^*$.

Show that $f(x, y)=(-1)^y x$ is an isomorphism from $\mathbb{R}^{+} \times \mathbb{Z}_2$ to $\mathbb{R}^*$. (REMARK: To combine elements of $\mathbb{R}^{+}\times \mathbb{Z}_2$, one multiplies first ...
Robair Garas's user avatar
4 votes
1 answer
78 views

Symmetry group of the unit circle in $\mathbb{R}^2$ versus $\mathbb{R}/\mathbb{Z}$.

I am studying a set of lecture notes on group theory, and I don't think I understand a point the author makes about the unit circle and its symmetry group in relation to $\mathbb{R}/\mathbb{Z}$. Let $...
JohnT's user avatar
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2 votes
0 answers
86 views

Isomorphism between two general linear group. [closed]

If $V$ is a vector space over the field $F$, the general linear group of $V$, written $GL(V)$, is the set of all bijective linear transformations $V\to V$, together with functional composition as ...
Tom's user avatar
  • 21
1 vote
0 answers
33 views

Number of conjugacy classes in a set of subgroups of $S_5$ which are isomorphic to klein's 4 group.

Let, K be a set of subgroups of S5 (symmetric group of 5 elements) that are isomorphic to the non-cyclic group of order 4. How many conjugacy classes are there in K? I know that a non-cyclic group of ...
Captain_Grothendieck 's user avatar
5 votes
1 answer
124 views

Is every group isomorphic to a set of isomorphisms?

Informally: Every group is representable (up to an isomorhism) as a group of isomorphisms. Formally: For every group $G$ there exists a binary relation $f$ on some set $U$ such that $G$ is isomorphic ...
porton's user avatar
  • 5,103
2 votes
1 answer
89 views

Special Isomorphism?

I was thinking about this concept the other day, but I couldn't reach a solid conclusion. Visualise your right hand in front of you, and put your thumb, second and third finger in a configuration such ...
J.D's user avatar
  • 1,279
4 votes
0 answers
119 views

How can I decide whether two groups defined by finite presentations are (or not) isomorphic?

I have the groups $G_1,G_2$ with presentations $$G_1 = \langle x,y : (y^2x)^2 = x^2, (x^2 y )^2 = y^{-2} \rangle = \langle x,y : x^{-1}y^2 x = y^{-2}, yx^2y^3 = x^{-2} \rangle \\ G_2 = \langle x,y : (...
Adrian's user avatar
  • 61
0 votes
0 answers
26 views

Strategy for Minimizing Open Rows in Matrix/Graph Traversal

I have a problem involving the traversal of a binary matrix (which can also be conceptualized as a graph traversal problem) under specific constraints, aiming to minimize the maximum number of ...
user2697423's user avatar
1 vote
1 answer
176 views

Subgroups of $\mathbb{Z}^n$ isomorphic to $\mathbb{Z}^n$

I am trying to prove the statement that all subgroups of $\mathbb{Z}^n$ isomorphic to $\mathbb{Z}^n$ are of the form $$b_1\mathbb{Z}e_1 \oplus b_2\mathbb{Z}e_2 \cdots \oplus b_n\mathbb{Z}e_n$$ where $...
Hanging Pawns's user avatar
4 votes
1 answer
155 views

Let $K\lhd G$ be s.t. both $K$ and $G/K$ are simple. Show that either $K$ is the only proper normal subgroup of $G$, or $G \cong K \times (G / K)$.

Sorry about the title, I couldn't fit the whole exercise (Exercise 8.1.6, Nicholson Introduction to Abstract Algebra 4th edition): Let $K \triangleleft G$ be such that both $K$ and $G/K$ are simple. ...
iwjueph94rgytbhr's user avatar
8 votes
1 answer
276 views

Existence of injective and surjective group homomorphisms in both directions implies existence of group isomorphism

Let $G$ and $H$ be groups such that there exist group homomorphisms $\iota_{HG}: G \hookrightarrow H$, $\iota_{GH}: H \hookrightarrow G$, $\pi_{HG}: G \twoheadrightarrow H$, $\pi_{GH}: H \...
Smiley1000's user avatar
  • 1,649
1 vote
1 answer
90 views

Defining a group isomorphism by mapping generators to generators

I am trying to understand how a group homomorphism mapping generators to generators can be shown to be an isomorphism. The example I have in mind is $S_3$ and $D_3$. The group presentations I have in ...
Valor Vaporeon's user avatar
3 votes
3 answers
164 views

Find the complete list of abelian groups of order $n$, up to isomorphism [duplicate]

Here is the question: Find the complete list of abelian groups of order $n$, up to isomorphism I don't understand what this is asking me. I've checked online, and I understand it's about the ...
Waterbloo's user avatar
  • 131
1 vote
0 answers
87 views

Show that as groups, $\mathrm{GL}(V)\cong\mathrm{GL}_n(\mathbb{F})$, with the isomorphism given by $\theta \mapsto A_\theta$

Show that as groups, $\mathrm{GL}(V)\cong\mathrm{GL}_n(\mathbb{F})$, with the isomorphism given by $\theta \mapsto A_\theta$. The notation can be found in my attempt. 1st Attempt: If $\dim_\mathbb{F} ...
Andrés de Fonollosa's user avatar
4 votes
2 answers
64 views

Example for $[G : G\cap H] \neq [H:G\cap H]$ with isomorphic subgroups $G \cong H$?

For isomorphic finite subgroups $G,H$ of a group $A$ it holds $[G : G\cap H] = [H: G\cap H]$, since $[G : G\cap H] = \frac{|G|}{|G\cap H|} = \frac{|H|}{|G\cap H|} =[H: G\cap H]$. Does this also hold ...
psl2Z's user avatar
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1 vote
1 answer
75 views

If $G \cong G'$ and $H \cong H'$ where $H \trianglelefteq G$ and $H' \trianglelefteq G'$, is then also $G/H \cong G'/H'$?

This seems pretty trivial, but I used it in an assignment and want to double-, triple check. If $G \cong G'$ and $H \cong H'$ where $H \trianglelefteq G$ and $H' \trianglelefteq G'$ are normal ...
soggycornflakes's user avatar
5 votes
1 answer
164 views

"Let $G$ be a finite group with a normal subgroup $N\cong S_3$. Show there is $H\le G$ s.t. $G=N\times H$." Does this $=$ really mean $\cong$?

I'm attempting Problem 60 on this problem set. Let $G$ be a finite group with a normal subgroup $N\cong S_3$. Show that there is a subgroup $H$ of $G$ such that $G = N \times H$. I'm guessing the ...
aleph2's user avatar
  • 934
1 vote
0 answers
54 views

Correspondence Theorem Finding Subgroups [closed]

Let $G$ be a group with a normal subgroup $N$ of order $7$ such that $G/N$ is isomorphic to the dihedral group of order $10$. Prove the following: (a) $G$ has a normal subgroup of order $35$. (b) $G$ ...
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