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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GAP Isomorphism between two fp p-groups [duplicate]
I have finitely presented nilpotent p-groups with exponenta 5 G = $\langle x1, x2, x3, x4: [x1, x2] = 1, [x3, x4] = 1, [x1, x4] = [x3, x2], [x2, x4] = [x1, x3]^{-2}] \rangle$ and H = $\langle x1, x2, ...
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Is $\mathbb{R}^4$ isomorphic to $\mathbb{C}^2$, $\mathbb{R}^4$ isomorphic to $\mathrm{Mat}_{4 \times 4}(\mathbb{R})$ as $\mathbb{R}$-vector spaces?
Is $\mathbb{R}^4$ isomorphic to $\mathbb{C}^2$ as $\mathbb{R}$-vector spaces?
Is $\mathbb{R}^4$ isomorphic to $\mathrm{Mat}_{4 \times 4}(\mathbb{R})$ as $\mathbb{R}$-vector spaces?
For the first one, ...
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Are Gl(2,R) and Sl(2,R) isomorphic? [duplicate]
Gl(2,R) and Sl(2,R) are both infinite groups, though Sl(2,R) is a proper subgroup of Gl(2,R). So is it possible to get such isomorphism
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Isomorphism between Cartesian and Polar coordinates
I have a homework problem that is explicitly stated as follows:
The set $C$ of the complex numbers can be defined in two different ways:
a. As a set of all ordered pairs $(x,y)$
where $x,y∈R$
b. As a ...
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Proving isomorphism between $\Bbb R^2$ and $\Bbb C$ ordered pairs
I have to establish an isomorphism between $\Bbb R^2$ and $\Bbb C$. Specifically, I have to establish an isomorphism between the ordered pairs $(x,y)$ and $(\phi,\rho)$, where $x,y \in \Bbb R^2$ and $\...
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Prove $GL_2(R)/S$ is isomorphic to $Z_2$ [duplicate]
I'm asked to show that $GL_2(\mathbb{R})/S$ is isomorphic to $Z_2$ where
$GL_2(\mathbb{R})=$$\left\{A=
\begin{pmatrix}
a & b\\
c & d
\end{pmatrix}, |a,b,c,d \in \mathbb{R} , det A \neq 0 ...
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What are the ingredients of the Sunada's theorem in this example?
I recall what this theorem says:
Let M be a Riemannian manifold upon which a finite group G acts by isometries; let H and K be subgroups of G that act freely. Suppose that
H and K are almost conjugate ...
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Ordered groups over the same underlying set and total order, can they be group isomorphic without being order isomorphic?
Given ordered groups $H:=(A,+)$ and $H':=(A, \boxplus)$ as per the same total order $ \leqslant$ of A,
can $H$ and $H'$ be group isomorphic without being order isomorphic?
I understand that there can ...
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The group $S_7$ has no subgroups of order $840$.
good afternoon everyone. I was studying group theory and came across the following exercise:
Let $x=(3,4,8,6)(5,7)$, $y=(2,1)(8,6)(5,7)(3,4)$, and $G=\langle x,y\rangle$.
Denote by $A,B,C$ and $D$ ...
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Isomorphic groups with $Sym(G/H)$ [duplicate]
Let $G$ a group.
If $H<G$, and $[G:H]=d$, so
$$ F: G\rightarrow Sym(G/H) $$
$$ g \mapsto F(g) $$
$$F(g): G/H \rightarrow G/H$$
$$ xH \mapsto g*xH $$
Prove that $Ker(F)= \displaystyle {\bigcap_{x \...
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External direct product realized as an internal direct product. What is the meaning of "realize"? Is my isomorphism correct?
I don't quite understand this sentence: "Every external direct product is naturally realized as an internal direct product."
Does "realize" mean that $H\times K$ is equal to $H+K$ (...
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Is $\mathbb(Q, +)$ isomorphic to $\mathbb(R, +)$? (without using uncountability)
I came across this problem:
Prove that $\mathbb(Q, +)$ is not isomorphic to $\mathbb(R, +)$ without using the argument of uncountability.
The alternative argument given was the following:
$\mathbb(Q, +...
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Decomposition of Homomorphism
Is it correct that if $\psi \ : G \rtimes_{\phi_1} H \to G \rtimes_{\phi_2} H$, then $\exists f \in Aut(G)\ and \ g \in Aut(H)$, such that $\psi(G \rtimes_{\phi_1} H) \cong f(G) \times g(H)$
What ...
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Can break the isomorphism between 2 semi-direct products componentwisely?
Consider the isomorphism $$\begin{align*}
\psi : G \rtimes_{\phi_{1}} H \to G \rtimes _{\phi_{2}} H
\end{align*}$$ where $G \rtimes_{\phi_{i}} H$ is the semi-direct product of $G$ and $H$ with ...
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How to prove $\mathbb{R}^{2} \rtimes_{\phi_{n}} SO(2) \not \cong \mathbb{R}^{2} \rtimes_{\phi_{m}} SO(2)$ for different $m,n \in \mathbb{N}$
Let $$\begin{align*}
r_{\theta} = \begin{pmatrix}
\cos\theta & -\sin \theta \\ \sin \theta & \cos \theta
\end{pmatrix}
\end{align*}$$ for $\theta \in[0,2\pi]$ . Then all $r_{\theta}$ forms $...
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Let $A,B, C,A',B'$ be an abelian groups. Let $f:A\to B$ and $g:B\to C$ be group homomorphisms.
Let $A,B, C,A',B'$ be an abelian groups.
Let $f:A\to B$ and $g:B\to C$ be group homomorphisms.
Suppose $A'\cong A$, $B'\cong B$ and $C\cong C'$ as abelian groups and $imf\subset kerg$.
Let $f':A'\to B'...
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Finding a non-abelian group with $54$ elements non-isomorphic to $D_{27}$
My goal is to find a non-abelian group with $54$ elements non-isomorphic to $D_{27}$ and for that matter, I tried $\mathbb{Z_3} \times D_9$, now we have that $Z(\mathbb{Z_3} \times D_9) =\mathbb{Z_3} \...
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Show that $G = HK$
Let $m, n$ be coprime positive integers. Let $G$ be an abelian group
of order $mn$. Let
$$
H = \{ g^m : g \in G \} \quad K = \{ g^n : g \in G \} \\
$$
Show that $G = HK \cong H \times K$
So far I've ...
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When is general linear group isomorphic to special linear group?
Can $\operatorname{GL}_n(\Bbb F_{q})$ be isomorphic to $\operatorname{SL}_m(\Bbb F_{r})$, where $\Bbb F_q$, $\Bbb F_r$ are finite fields with $q,r$ elements respectively?
By considering the center, $$\...
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Prove $k$ and $|G|$ is coprime when $\varphi_k(g)=g^k$ is a automorphism
The proposition is as follows:" Let $G$ be a finite Abelian group. Prove that $\varphi_k(g) = g^k$ is an endomorphism of $G$. If $\varphi_k$ is an automorphism of $G$, then $k$ and $|G|$ are ...
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Index of the multiple of a group in itself
Let $m, n \geq 1$, prove that $[\mathbb{Z}/m \mathbb{Z} : n\left( \mathbb{Z}/m \mathbb{Z} \right)] = \text{gcd}(m, n)$
When writing out the index as the cardinality of the quotient space, this looks ...
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Is $\mathbb{Q^{*}}$ under multiplication isomorphic to $\mathbb{Q^{+}}$ under multiplication? [duplicate]
I need to prove or disprove that $\mathbb{Q^{*}}$ under multiplication is isomorphic to $\mathbb{Q^{+}}$ under multiplication. However, I am not entirely sure if my proof is correct.
Note that $\...
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Cantor-Schröder-Bernstein theorem for finitely generated abelian groups [duplicate]
I study theory of groupa now. I tried but I can't solve this problem:
Let $A$ and $B$ be finitely generated Abelian groups, and each one of them is isomorphic to a subgroup of the other. Prove that $...
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The projective linear group $PGL_2(q)$ and $SL_2(q)$ [duplicate]
I am trying to show that for $q$ odd prime number, the groups $SL_2(q)$ and $PGL_2(q)$ are not isomorphic. On a similar post:Is $PGL_2(q)$ isomorphic to $SL_2(q)$ . Stephan and Andres stated that ...
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Is there any spin group $Spin(n)$ for any value of $n$ which is isomorphic to $SU(3)$?
I was wondering about the spin group and its relations with the unitary groups.
Like we have $Spin(2)\simeq U(1)$, $Spin(3) \simeq SU(2)$.
I was wondering whether this relation can be taken further to ...
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Similar short exact sequences where the first abelian group is different
I am trying to get a better understanding of short exact sequences of abelian groups. I know that if
$0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$
and
$0 \rightarrow A' \rightarrow B' \...
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Show there is no isomorphism between a subgroup of $ \mathbb{Q}$ and $\mathbb{Z} \times \mathbb{Z}$ [duplicate]
I am currently working on a problem trying to prove that there is no subgroup $ H $ of $\mathbb{Q}$ such that $H \cong \mathbb{Z} \times \mathbb{Z}$.
I was able to show that $\mathbb{Q}$ cannot be ...
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1
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Infinite group without a subgroup isomorphic to $\mathbb{Z}$?
I am looking for examples of infinite groups in which no subgroup is isomorphic to $\mathbb{Z}$. Some of the examples I can think of are: $$\prod_{i=1}^\infty \mathbb{Z}_2, \quad \bigcup_{i=1}^\infty ...
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Show $f(x)=1/x$ and $g(x)=(x-1)/x$ produce a group of functions isomorphic to the symmetric group $S_3$ when binary operation is used as map.
I found this question Isomorphisms between group of functions and $S_3$ but they don't show why $\phi$ is an isomorphism.
I found that
$$f(x) = 1/x$$
$$g(x) = (x-1)/x$$
$$f(f(x)) = x$$
$$f(g(x)) = x/(...
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$H,K \leq G$, $|H| = 126, |K| = 228$. Show that $H \cap K \leq G$ is either cyclic or isomorphic to $D_3$
So, I don't have to prove that $H \cap K$ is a subgroup of $G$, and $G$ is finite.
I know that any group of order $6$ is isomorphic to either $D_3$ or $\mathbb{Z}_6$ (which is cyclic) so I'm assuming ...
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Must two groups with all subgroups of the same order be isomorphic? [duplicate]
Let $G,H$ be finite groups such that there's bijection $\varphi$ from the subgroups of $G$ into the subgroups of $H$ satisfying:
$|g_1|=|\varphi(g_1)|$;
$g_1 < g_2$ if and only if $\varphi(g_1)&...
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Proving there is a unique subgroup $H\le S_4$ isomorphic to $G=\langle (1~~3),(2~~3~~4)\rangle\le S_7.$
Suppose $G$ is a subgroup of $S_7$ generated by the elements $a=(1~~3)$ and $b=(2~~3~~4).$ Prove there exists a unique subgroup $H$ of $S_4$ isomorphic to $G.$
My attempt
Both $a$ and $b$ fix the set ...
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For group $G,H$, $f: G \to H$ be an homomorphism, and $N$ be a normal subgroup of $G$, $\frac{G}{N} \cong Im(f) \implies N \cong Ker(f)?$. [duplicate]
I am a complete beginner to Abstract Algebra. Today I asked my professor the following:
For group $G$ and $H$, let $f: G \to H$ be an homomorphism, and $N$ be a normal subgroup of $G$, if $\frac{G}{N}...
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1
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Character tables of isomorphic groups
Let us consider two groups $G$ and $G'$ which are isomorphic to each other. Since isomorphic groups can be considered as same upto isomorphism, is the character table same for both groups?
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Let $h\in G$. Let $\phi_h : G \to G$ st $g\mapsto hgh^{−1}$. Then $G$ is abelian iff $\phi_h = \operatorname{id}_G$ for all $h\in G$. [closed]
Let $G$ be a group and $h\in G$. Consider the map $\phi_h : G \to G$ st $g\mapsto hgh^{−1}$, $G$ is an abelian group if and only if $\phi_h = \operatorname{id}_G$ for all $h\in G$.
I know what it ...
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If $(G,\cdot)$ and $(G,+)$ are groups, are they always isomorphic?
It is possible for a group to have alternative operation. For example, consider the following operations on $\mathbb{Z}^2$: $(x_1,y_1)\cdot(x_2,y_2)=(x_1+x_2,y_1+y_2+x_1x_2)$ and $(x_1,y_1)+(x_2,y_2)=(...
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$U(24)$: internal direct product and generators of a subgroup of a full symmetric group
I'm being confused by the following question:
Build a permutation representation of U(24). List a representation of each element. Then, construct the group as a subgroup of a full symmetric group ...
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3
votes
3
answers
79
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Rotations of a cube isomorphic to $S_4$
I'm trying to prove that the group $G$ of rotations of the cube is isomorphic to $S_4$. I have a very rough sketch of a proof, which has the following three components:
$G$ has $24$ elements
There ...
0
votes
1
answer
68
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About using onto and into in the definition of automorphism
Definition. An isomorphism $\phi: G \to G$ of a group $G$ with itself is an automorphism of G.
Why is it that rephrasing this definition with "An automorphism of a group G is an isomorphism ...
user1221612
2
votes
1
answer
70
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Quotienting the unit circle by $\{\pm 1\}$
I'm trying to solve the following problem.
Let $S^1$ denote the set of $z \in \mathbb{C}^{*}$ with modulus $1$ and $H = \{\pm 1\}$ a subgroup. Prove that $S^1/H \cong S^1$.
I'm a bit stumped with ...
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1
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Can anyone give me an example of a map between two groups that is homomorphic but not isomorphic?
I'm learning homomorphism and isomorphism in basic group theory. Can anyone give me an example of a map between two groups that is homomorphic but not isomorphic? I couldn't think of one.
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votes
1
answer
109
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Quotient group by two isomorphic groups
In the processes of studying some questions I suddenly realized something basic but weird. We have $\mathbb{Z}\cong 2\mathbb{Z}$, but $\mathbb{Z}/\mathbb{Z}\ncong \mathbb{Z}/2\mathbb{Z}$. It looks ...
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3
votes
1
answer
101
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Complement for the Center of a Group
On P. 21 of Keith Conrad's notes on semidirect products, he made the following claim:
For a group $G$, let $H=Z(G)=Z$. ... Conversely, if $G\cong Z\times G/Z$ (any isomorphism at all), then $Z$ has a ...
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votes
1
answer
62
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Free product and direct product of $\mathbb{Z}$
I'm studying Seifert Van Kampen theorem and I evaluate the fundamental group of torus seen as a quotient space in particular
$$T = [0,1]\times[0,1]/\sim\\
(x,0)\sim(x,1) \\
(0,y)\sim(1,y)$$
With $\pi:[...
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vote
4
answers
89
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Is $\mathbb{Z}_4 \simeq U(5)$? [duplicate]
I am doing a tutorial question where it ask whether or not the groups of order 4 $(\mathbb{Z}_4,+\mod 4)$ and $(U(5),\times _5)$ are isomorphic to each other.
I drew out the Cayley tables and got
$$
\...
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Let $G$ be a group isomorphic to $Z_{n_1} \oplus Z_{n_2} \oplus \cdots\oplus Z_{n_k}$.
Gallian's "Contemporary Abstract Algebra", Chapter 8 Problem 44:
Let $G$ be a group isomorphic to $Z_{n_1} \oplus Z_{n_2} \oplus \cdots \oplus Z_{n_k}$. Let $x$ be the
product of all ...
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Is this group isomorphic to the real numbers?
I have a locally compact abelian group $G$ with the following properties:
It is connected (therefore divisible) and non-compact;
It admits a $\mathbb{Q}$-vector space structure and for any $g \neq 0$,...
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1
answer
66
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Image of a group element under isomorphism
I'm not completely sure if this proof works that an isomorphism preserves the order of elements. Here is my attempt at a proof.
Let $f: G \to H$ be an isomorphism of groups and $x \in G$. Then
$$\...
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4
votes
6
answers
492
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Bijective group homomorphisms are isomorphisms [closed]
Why are all bijective group homomorphisms $\phi: G \to H$ automatically isomorphisms? That is, their inverses are also homomorphisms.
I have seen this fact mentioned before, but never a proof.
We ...
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2
answers
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Orbit stabiliser theorem as an analogue to first isomorphism theorem
The notes I'm using to study group theory make a remark that another appropriate name for the "orbit stabiliser theorem" is the "first isomorphism theorem for group actions". For ...
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