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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How to show that two groups are not isomorphic. [duplicate]

I have learned various theorems that tell me when two groups are isomorphic. For example, if the greatest common divisor of $j$ and $k$ is equal to one, then $\mathbb{Z}_j\oplus\mathbb{Z}_k\cong\...
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Is $G\otimes\mathbb Z^X$ isomorphic to $G^X$?

Let $G$ be an abelian group and let $X$ be a finite set. For a given group $T$, let $T^X$ be the group of functions $X\to T$ where the operation is defined pointwise. Is it true that $G\otimes\mathbb ...
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Proof idea: $S_3 \cong D_3$ [duplicate]

I am trying to show that $S_3$ is isomorphic to $D_3$ as groups. The definitions I'm working with are $$ S_3 = \left \langle (12), (123) \right \rangle, \; D_3 = \left \langle r, s \mid r^3 = s^2 = 1, ...
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Show that $SL_2(F_3)/Z(SL_2(F_3)) \cong A_4$

Show that $SL_2(F_3)/Z(SL_2(F_3)) \cong A_4$ I know that $|SL_2(F_3)/Z(SL_2(F_3))|= 12$. Then if the quotient group has a normal subgroup of order $4$ then it is isomorphic to $A_4$. Suppose that it ...
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Example of $H$, $N$ two normal subgroups of a group G such that $G/N \cong G/H$ but $N \ncong H$ [duplicate]

I have tried to prove that if $H$, $N$ are two normal subgroups of a group G such that $G/N \cong G/H$, then $N \cong H$. I think that it is not possible, but I can't find a counterexample. What is an ...
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Isomorphism between group mod inclusion and cartesian product mod normal subgroup

Given $G_1, G_2$ Groups and $N$ a normal subgroup of $G_1\times G_2$, I have already proved that the projections $p_i: G_1\times G_2\to G_i$ and $i_1:g_1\mapsto (g_1,1), \; i_2:g_2\mapsto (1, g_2)$ ...
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For every abelian group $A$, is $n A / m A$ isomorphic to $A / (m - n) A$?

Let $A$ be an abelian group. For arbitrary positive integer $n$, $m$, I think $nA / mA$ and $A / (m-n)A$ are isomorphic as abelian groups. $$ nx \;\mathrm{mod}\, mA \mapsto x \;\mathrm{mod}(m-n)\, A ...
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Order of automorphisms of non-abelian group

I'm trying to solve the problem, but it doesn't work. Please help me to solve it. Problem: prove that $\bigl| \operatorname{Aut} \, (G) \bigr| \ge 8$ if $G$ is non-abelian and not isomorphic with $S_3$...
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Show that the intersection of these two subgroups has prime index

Let the subgroup $N$ be normal in the finite group $G$ with index a prime $p$. Let $H$ be a subgroup of $G$ which is not contained in $N$. I would like to show that $H \cap N$ is normal in $H$ with ...
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Prove that if $H_1\lhd G, H_2\lhd G, H_1H_2=G, H_1\cap H_2=\{1\}$, then $G/H_1\simeq H_2$.

Problem: Suppose $G$ is a group, and $H_1, H_2$ are subgroups of $G$. Prove that if $H_1, H_2$ are normal subgroups of $G$, $H_1H_2=\{xy: x\in H_1, y\in H_2\}=G$, and $H_1\cap H_2=\{1\}$ where $1$ is ...
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Verifying the quotient groups of matrices from an isomorphism if the matrices are equivalent under row&column operations.

The question is as followed: Consider $A,B \in M_{m\times n}(\Bbb Z)$. Show that there are $P \in GL_{m}(\Bbb Z)$ and $Q \in GL_{n}(\Bbb Z)$ such that $PAQ = B$ if and only if abelian groups $\Bbb Z^{...
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Cayley's theorem - How many cycles of each type exists

Suppose we have a group isomorphism $G \cong H \leq S_{q+1} $. Let $ \chi_{q+1} $ denote the number of $(q+1)$ - cycles $\in H$ and suppose $q$ is an odd prime, then I've shown that $ \frac{q-1}{2} \...
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What is the example of non canonical isom? [closed]

In mathematics, isomorphism $f$ between two objects are sometimes called canonical when $f$ is unique as a map, in other words, everyone choose the same map, no different map gives the isomorphism. ...
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Isomorphisms preserve order: the infinite case

Given a group $G$ and a isomorphism $\varphi: G \to H$, I know how to prove that $|x| = |\varphi(x)|$ in the finite case. If $|x| = n$, I prove that $\varphi(x^n) = \varphi(x)^n$, and since $x^n = e$, ...
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Groups of the form $(\Bbb Z_{13} \times \Bbb Z_7)\rtimes \Bbb Z_3$

Consider the semi-direct product: $(\Bbb Z_{13} \times \Bbb Z_7)\rtimes \Bbb Z_3$ To construct a group $G$, we need homomorphisms $\theta$: $\Bbb Z_3 \rightarrow \text{Aut}(\Bbb Z_7)$ and $\theta_2$: $...
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Are two matrices isomorphic? (as rings and as group) [closed]

Assume that $M_2(R) , M_3(R)$ are matrices with real cells $2 \times 2$ , $3 \times 3$ respectively. 1)Are $M_2(R) , M_3(R)$ Isomorphic as rings under addition and multiplication ? why? 2) Are $M_2(R) ...
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Find $d$ such that $(M_d, \cdot)$ group is isomorphic to the $(\mathbb{C}^*, \cdot)$?

Let $d$ be an arbitrary real number and $M_d=\left\{ \begin{pmatrix} a& db\\ b& a \end{pmatrix}\in\mathcal{M}_2(\mathbb{R}), \text{where } a^2-db^2\neq 0\right\}. $ The problem is: show that $...
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Prove that there exists finitely many groups of order n.

Every finite group is isomorphic to some permutation group. Any group of order $n$ can be embedded into $S_n$. (We say that group $G_1$ is embedded into $G_2$ if there is $f:G_1\to G_2$ that is ...
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Prove or disprove: there exists no binary operation $*$ on $\mathbb Q^+$ s.t. $(\mathbb Q^+,*)$ is not isomorphic to $(\mathbb Q^+,×)$

I am a first year college student who just started studying abstract algebra. I have been discussing the following problem with my friends at another university for a couple of days: Prove or disprove:...
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Intuition for finding a group $G$ such that $G \cong \mathrm{Aut}(G)$

I'm trying to find a group such that the map $G \to \mathrm{Aut}(G)$ sending $a$ to the the conjugation map $\phi_a (x) = axa^{-1}$ is an isomorphism. I know that $G = S_3$ works and I know how to ...
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1 vote
1 answer
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Check if kernel and image for dihedral group defined correctly

Suppose that $n = dm$ where $d$ and $m$ are positive integers with $m\ge 3$. Consider the dihedral group $D_n = \langle \{\mu, \rho\}\rangle,$ where $|\mu| = 2$, $|\rho| = n$ and $\rho\mu = \mu\rho^{−...
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Find specific example for where the case works (topic: isomorphism and normal subgroups)

Consider groups $G_i, K_i, F_i$ such that $K_i \triangleleft G_i$ and $G_i / K_i\cong F_i$ , for $i = 1, 2$. In each case, find an example with the given properties or prove that no such example ...
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question for group theory [duplicate]

hello i am korean student of mathematical education. first of all, i am sorry for lack of my english skills. here is question. suppose H, K is normal subgroup of group G. If H is isomorphic to K, is ...
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Classifying groups of order less than or equal to $4$ up to isomorphism

I'm trying to prove the following assertions. Classify groups of order less than or equal to $4$ along the following lines: (a) There is one group of order $2$ up to isomorphism. (b) There is one ...
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Question about isomorphism between the subgroups $N$ and $a^{-1}Na$

In showing isomorphism between the subgroups $N$ and $a^{-1}Na$ of a group $G$, one usually define a function $f:N\rightarrow a^{-1}Na$ by $f(n)=a^{-1}na$ for all $a\in G$ and some $n\in N$. This ...
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Why does injectivity imply to $|G/(H \cap K)|\leq |G/H|\cdot|G/K|?$

I'm looking through some proof about the inequality in the title, the one defines: $$\phi: G/(H \cap K)\rightarrow G/H\times G/K$$ $$\phi(g(H \cap K))=(gH,gK)$$ Note that $\phi$ is injective, I'd like ...
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2 votes
0 answers
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Finding isomorphisms among groups

The lecture notes I'm working through pose an exercise to find groups that are isomorphic to each other or to subgroups of other groups. The groups listed so far are: The trivial group $\{e\}$ ...
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-1 votes
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Let a and b be nonzero rational numbers. Prove that Q(√a) is isomorphic to Q(√b) if and only if $a/b=q^2$ for some q∈Q. [closed]

Let $a$ and $b$ be nonzero rational numbers. Prove that $\Bbb{Q} (\sqrt a)$ is isomorphic to $\Bbb{Q} (\sqrt b)$ if and only if $a/b= q^2$ for some $q ∈\Bbb{Q}$. I have the hint that If $\sqrt a∈\Bbb{...
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On isomorphic normal subgroups of a group

Well this is quite stupid a question because the intuition is false. See comments and answers below. Original question: Consider $H_1,H_2\le G$, $H_1 \cong H_2$. It seems trivial that if $H_1 \unlhd ...
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Quotient group by normal closure of union

Let $G$ be a group and $H, N \subseteq G$ be subsets of the group. Let $\overline{A}$ denote the normal closure of any subset $A\subseteq G'$ in some group $G'$. Let $\pi: G \to G/\overline{N}$ denote ...
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The given function $\phi$ is not an isomorphism because it is not surjective. But it's not injective either, right?

Here's the problem I did for homework from A First Course in Abstract Algebra, 7th Edition by John B. Fraleigh. I just want to check if my reasoning is correct on problem number 15 from Section 3: ...
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8 votes
1 answer
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Do the groups $\operatorname{SL}$, $\operatorname{PGL}$, and $\operatorname{PSL}$ over a field $K$ always have the same automorphism group?

Let $K$ be a field, then $\operatorname{GL}(n,K)$ consists of the $n\times n$ invertible matrices, $\operatorname{SL}(n,K)$ consists of the $n\times n-$matrices with determinant $1$, and $\...
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4 votes
1 answer
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Show the function between the dihedral groups is well defined

Suppose that $n = dm$ where $d$ and $m$ are positive integers with $m\ge 3$. Consider the dihedral group $D_n = \langle \{\mu, \rho\}\rangle,$ where $|\mu| = 2$, $|\rho| = n$ and $\rho\mu = \mu\rho^{−...
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Galois group $G$ where every element fixes a root is trivial

Let $K$ be the splitting field of a separable irreducible polynomial $f(x) \in F[x]$ of degree $n$ and let $G = Gal(K/F)$. If for each $g \in G$, there is a root $\alpha$ of $f$ such that $g(\alpha) =...
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1 answer
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Do isomorphic groups have the same number of Sylow $p$-subgroups?

I think that isomorphic groups should have the same number of Sylow $p$-groups, but I am not sure why, I am a little stuck on this, I really don't know where to even begin, or if this is even true (...
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2 votes
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When is $D_n\approx\operatorname{Aut}(D_n)$? [duplicate]

We define the group $D_n$ to be the dihedral group of order $2n$ (equivalent to the group of rotations and reflections on a regular $n$-gon) and $\operatorname{Aut}(G)$ to be the group of ...
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Is $\operatorname{Hol}(D_4)$ isomorphic to a familiar group?

We define the holomorph of a group, $\operatorname{Hol}(G)$, as its semidirect product $G\rtimes _f\operatorname{Aut}(G)$. As it happens (as is shown here), $D_4\approx\operatorname{Aut}(D_4)$, and we ...
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Proof verification of Dummit and Foote exercise 1.6.24

Problem 1.6.24 in D&F: Let $G$ be a finite group and let $x$ and $y$ be distinct elements of order $2$ in $G$ that generate $G$. Prove that $G \cong D_{2n}$, where $n = |xy|$. My proof: If $t = ...
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What is wrong with my proof of group order?

Let $\phi: G\rightarrow H$ be an isomorphism of groups and let $a \in G$ be of order $n$. Show that the order of $\phi(a)$ is also $n$. I was given this problem a week ago during a quiz and my ...
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16 votes
1 answer
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If $G,H$ are finite groups, then $G\times G\cong H \times H$ implies $G \cong H$

Proposition. Let $G,H$ be finite groups (abelian or not). Then the following implication holds: $$G\times G\cong H\times H \Rightarrow G\cong H.$$ In the case of $G,H$ both abelian, one can use the ...
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1 vote
2 answers
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Prove that ${\rm Inn}(S_n)$ isomorphic to $S_n.$

Show that ${\rm Inn}(S_n)$ isomorphic to $S_n$ for $n\ge3$. To do this, if I define some isomorphic function say $\phi$, where $\phi: S_n \to{\rm Inn}(S_n)$, then show that $\phi$ is bijective (by ...
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Isomorphism Between Two Subgroups of $GL_2(\mathbb{F})$

Good day! I am struggling with the following question in my group theory class for quite some time and would love to recieve a hint: Let $\mathbb{F}$ denote some field and $\mathbb{F^*} = \mathbb{F} \...
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3 votes
1 answer
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Can the direct product of nontrivial groups $A\times B$ ever be isomorphic to a free group?

Can the direct product of nontrivial groups $A\times B$ ever be isomorphic to a free group? Let's say that for two groups $A$ and $B$ we have that $A\times B \cong F_n$ where $F_n$ is the free group ...
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1 vote
1 answer
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Why is $(\mathbb{Z}\oplus \mathbb{Z}\oplus \mathbb{Z})/(a_1+a_2+a_3=0)\cong \mathbb{Z}$?

Why is $(\mathbb{Z}\oplus \mathbb{Z}\oplus \mathbb{Z})/(a_1+a_2+a_3=0)\cong \mathbb{Z}$? That is: What would be the isomorphism to see this? And, in general, is there a way to find an isomorphism $G/N\...
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2 votes
0 answers
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A question on group isomorphism between $(\Bbb{R},+) $ and $(\Bbb{C}, +) $ .

$(\Bbb{R},+) $ and $(\Bbb{C}, +) $ are group-isomorphic. Consider, two vector space $\Bbb{R}_{\Bbb{Q}}$ and $\Bbb{C}_{\Bbb{Q}}$. Any two vector space over the same field having same dimension are ...
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1 vote
1 answer
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Let $G=D_8\times\mathbb{Z_6}$ and $N=D_8\times\{0\}$. Prove that $G/N$ is isomorphic to $\mathbb{Z}_6$.

Let $G=D_8\times\mathbb{Z_6}$ and $N=D_8\times\{0\}$. Prove that $G/N$ is isomorphic to $\mathbb{Z}_6$. My attempt: I proved that $N$ is a normal subgroup of $G$. Then we can define $$G/N = \{gN \mid ...
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If $(\mathbb{Z}, +) \cong (C_\infty, \cdot)$, then $\exists K$ such that $K \trianglelefteq C_\infty$ and $K \cong 2\mathbb{Z}$?

If $(\mathbb{Z}, +) \cong (C_\infty, \cdot)$, then does there exists a normal subgroup from $C_\infty$ (call it $K$) such that $K \cong 2\mathbb{Z}$? I tried the following: If $q \in C_\infty$, then $\...
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0 votes
1 answer
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Isomorphism of $\mathbb{Z}_{30}$ to $ \mathbb{Z}_2\times \mathbb{Z}_3\times \mathbb{Z}_5$

Hello I have a difficulty with this exercise the idea is to show $$\begin{align} \phi: \mathbb{Z}_{30} &\to \mathbb{Z}_2\times \mathbb{Z}_3\times \mathbb{Z}_5 \\ [x]_{30} &\mapsto ([x]_2,[...
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Prove that $\hspace{1mm} (\mathbb{R}^{n+1} \setminus\{0\})/{]0, \infty[}\cong\mathbb{S}^n$

I have been struggling to get a better grasp of the overall picture of what's going on throughout my solving process, perhaps, it's due a lack of a better understanding of some of the concepts ...
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1 answer
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Prove or disprove : i) $(\mathbf{S},+,.)\cong (\mathbf{R},+,.)$ (as rings) ii) $(\mathbf{S},+)\cong (\mathbf{R},+)$ (as groups)

Let $\mathbf{S}=\left\{\left[\begin{array}{ll}\mathbf{a} & \mathbf{b} \\ 0 & \mathbf{a}\end{array}\right]: \mathbf{a}, \mathbf{b} \in \mathbb{Z}_{2}\right\}$, $\mathbf{R}=\mathbb{Z}_{2} \times ...
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