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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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Center is a normal subgroup of G

This is a problem from Herstein's Topics in Algebra. I have already shown the above result using the definition of normal subgroup. But now I want to prove it by constructing a homomorphism such ...
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1answer
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bijection between a set and a set of functions

Question: For any function $f : A → B$ define an explicit isomorphism between A and the graph $Γ_f ⊂ A×B$ the subset defined by the property that for each $a ∈ A$ there is exactly one pair $(a, b) ∈ ...
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Could $\langle \Gamma | R \rangle \cong \langle \Gamma | S\rangle$ if $\langle R\rangle \subsetneq \langle S\rangle$?

If we have two finitely presented groups $\langle \Gamma | R\rangle$ and $\langle \Gamma | S\rangle$ with $\langle R\rangle \subsetneq \langle S\rangle$, could they be isomorphic?
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Third Isomorphism Theorem statement

If $H \vartriangleleft G$ and $K \vartriangleleft G$ then $K/H \vartriangleleft G/H$. Then : $(G/H) / (K/H)$ is isomorphic to $G/H$ . I know this is the statement of the theorem but would it be ...
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isomorphism classes/number of models up to iso

What does it mean here on the page $73$ the second $\aleph_2$ in $\dot{I}(\aleph_2,\aleph_2)$ ?
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Can we quotient a set $A$ by another set $B$ such that $B\not\subset A$?

I'm reading Ash's Basic Abstract Algebra. Here: I'm a little bit confused. When we restrict the map $\pi: G\to G/N$, we get the map $\pi_0:H\to H/(H\cap N)$, right? Assuming this is true, I don't ...
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How similar are permutation groups that are isomorphic as abstract groups?

Let's say that two permutation groups $P_1$ and $P_2$ are isomorphic as abstract groups, but not necessarily permutation isomorphic. How similar will $P_1$ and $P_2$ be, and how much structure will ...
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For which $n\in \Bbb N$ is $H_n:=\{\alpha^2\mid \alpha\in S_n\}\cong A_n?$

I'm reading "Contemporary Abstract Algebra," by Gallian. This is inspired by Exercise 5.73 and Exercise 5.74 ibid. I have a preference for answers using only the tools available in the textbook so ...
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Do the octonions contain infinitely many copies of the quaternions?

Note: by "infinitely many", I'm confident I always mean $\beth_1$ many herein. We can easily show the quaternions contain infinitely many copies of $\Bbb C$ because, given any unit vector $\in\Bbb R^...
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1answer
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Show that $f^{-1}\{f(a)\}=a\text{Ker}f$

Let $G,G'$ be groups and assume $f: \space G \longrightarrow G'$ is a homomorphism. Show that $\forall f(a) \in \ \operatorname{Im}f$: $$ f^{-1}\{f(a)\}=a \operatorname{Ker}f $$ To begin with, the ...
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Show that $G'$ contains a subgroup of order $d$

Assume $G$ is isomorphic to $G'$ and $H$ is a subgroup of $G$ with $|H|=d$. Show that there exists $H'\leq G'$ s.t. $|H'|=d$. Let $f: G \longrightarrow G'$ be an isomorphism. Then, there ...
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The groups $G_1, G_2$ and the normal subgroups $N_1,N_2$ are isomorphic. Dis/prove $G_1/N_1\cong G_2/N_2$ [duplicate]

Let $G_1, G_2$ be groups, and let $N_1\trianglelefteq G_1, N_2\trianglelefteq G_2$ such that $G_1\cong G_2, N_1\cong N_2$. Prove or disprove $G_1/N_1\cong G_2/N_2$. Let $\varphi:G_1\to G_2, \psi:N_1\...
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Constructing an isomorphism of group products

To introduce some symbolism for clarity's sake, here is the problem I'm faced with in an assignment: Find an example of four groups $A,B,C,D$ such that $$A \times B \cong C \times D$$ but at ...
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Why is $Z^n/AZ^n \cong Z/d_1Z\oplus Z/d_2Z..$ where $d_i$ are the entries of normal form of A?

In trying to show that $Card(Z^n/AZ^n)=det(A)$ which has been answered earlier here $\mathrm{card}(\mathbb{Z}^n/M\mathbb{Z}^n) = |\det(M)|$? , the answer refers to the isomorphism $Z^n/AZ^n \cong Z/...
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1answer
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Existence and construction of isomorphism between finite groups

Assume I have two finite groups $G$ and $H$ of equal order. Further assume I have found minimal generating sets $A$ and $B$ for a the two groups respectively (of equal size) and additionally (see ...
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2answers
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Let $P$ be the group of strictly positive real numbers under multiplication. Prove that $P$ isomorphic to $(\mathbb{R}, +)$ [closed]

Let $P$ be the group of strictly positive real numbers under multiplication. Prove that $P$ isomorphic to $(\mathbb{R}, +)$
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Find group isomorphic to the quotient group

I'm working on a few problems of a similar type. I'm pretty confident I got the right answer but I'm not sure how to provide satisfiable enough proof that the answer is correct. I'd be glad if you ...
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3answers
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Prove or disprove. Let $G$ be an abelian group. Let $N \triangleleft G$. Then, $ G\cong N \times G/N. $ [duplicate]

Prove or disprove. Let $G$ be an abelian group. Let $N \triangleleft G$. Then, $$ G\cong N \times G/N. $$ I tried to prove this claim, but then it seems that since $G$ is abelian then every ...
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Normal subgroup and corresponding homomorphism

Topics in Algebra, a book written by Herstein, said the following thing on the first isomorphism theory: [The book says] Theorem 2.7.1 is important, for it tells us precisely what groups can be ...
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1answer
59 views

Which group is this quotient group isomorphic to?

Let $$G = \left\{\begin{pmatrix} q&0\\a+bi&q\end{pmatrix} \mid q \in \mathbb{Q}^\ast, a,b\in\mathbb{R}\right\}$$ and $$H = \left\{\begin{pmatrix} q&0\\a+ai&q\end{pmatrix} \mid q \in \...
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Isomorphism in abelian group [duplicate]

Prove that if $G$ is a finite abelian group and $H$ is a subgroup of $G$ then $G$ contains a subgroup isomorphic to $G/H$
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Proving $(\Bbb R,+)$ has no proper subgroup isomorphic to itself

A captioned image with the text "My love for you is like a group which has a proper subgroup isomorphic to itself" was recently posted in a group chat I'm in. Ignoring the argument about the ...
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Find all non-isomorphic abelian groups s.t. $|G| \leq 30$ and $g^{12}=1, \, \forall g\in G$

Assume the prime factorization of the order of $G$: $$ |G|=p_1^{a_1}p_2^{a_2}\dots p_r^{a_r} $$ The condition $$ g^{12}=1,\, \forall g\in G\tag{1} $$ in other words means that we must find those ...
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1answer
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Are given groups isomorphic

Are there isomorphic groups between $\Bbb{A}_4\times \Bbb{Z}_3,\Bbb{D}_{18},\Bbb{D}_{9}\times \Bbb{Z}_2,\Bbb{S}_3\times \Bbb{S}_3$? Where $\Bbb{D}$ is the dihedral group,$\Bbb{A}$ alternating group ...
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Banach algebra $l^p$ is not isomorphic to $C^{*}$ algebra

Consider commutative Banach algebra $l^p$, $p \in [1,\infty)$ with multiplication by coordinates. I know, that $\Delta (l^{p})=\{e_n : n \in \mathbb{N}\}$ - set of canonical functionals. We know that $...
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1answer
35 views

Banach Algebra Isomorphism

Let $X=l^1$ with coordinate multiplication - it is commutative Banach algebra without unit. The Gelfand transformation is defined as $\widehat{x}(e_n)=x_n$ for $x \in l^1$. I would like to prove, that ...
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$f: G \to \mathbb{C}^*$ is a homomorphism. Show that the sum $\sum f (g) = 0$ or $n$

Let $ \mathbb{C}^*$ be the multiplicative group of non-zero complex numbers. Let $G$ be an abelian group and suppose $f: G \to \mathbb{C}^*$ is a homomorphism. Prove that $\sum_{g \in G} f(g)=n$ or, $\...
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2answers
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Prove that $X/A \cong (\Bbb{R^*};.)$.

Let $$\begin{align} X &=\left\{r\left(\cos \dfrac{k\pi}{3}+i \sin \dfrac{k\pi}{3}\right): r \in \Bbb{R^*},k \in \Bbb{Z}\right\}, \\ A &=\{z \in \Bbb{C}|z^3=1\}. \end{align}$$ We can show ...
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If $P$ is a Sylow $p$-subgroup of $G$ then there exists a normal subgroup $K$ such that $G/K$ is isomorphic to $P/P \cap G'$

In 'Finite Groups' by Gorenstein, it is stated that if $P$ is a Sylow $p$-subgroup of $G$ then there exists a normal subgroup $K$ such that $G/K$ is isomorphic to $P/P \cap G'$′. The proof is the ...
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There exists an element of order $5$ in $G$

Let $G$ be a finite group and assume $\varphi: \, G \longrightarrow \mathbb{Z}_{10}$ is a group epimorphism. I want to show that there exist an $a \in G$ s.t. $\text{ord}(a)=5$. At first, the cyclic ...
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What's the point of studying isomorphisms?

I'm trying to learn about groups/rings and the concept of isomorphisms appears everywhere. I understand that an isomorphism between groups/rings shows that arithmetic in both structures is essentially ...
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3answers
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If $\varphi$ is a homomorphism, $\text{ord}(\varphi(a))$ divides $\text{ord}(a)$

Assume $\varphi: \, G \longrightarrow G'$ is a homomorphism and $a \in G$ an element of finite order. If $m=\text{ord}(a)$ and $n=\text{ord}(\varphi(a))$, show that $n \big| m$. To begin with, it's ...
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Existence of abelian group which has no “square-root” but whose “cube” has a “square-root”

Does there exist an abelian group $G$ such that $G \ncong H \times H$ for every abelian group $H$ but $G \times G \times G \cong K \times K$ for some abelian group $K$ ? Also see Existence of ...
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1answer
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Is $U_8$ isomorphic to $K_4$ ( Klein Group)

$U_8=1,3,5,7$ since this group has one element of order one, three elements of two order and no element of $4$ order .. so does the Klein group. Both $U(8)$ and the Klein group are non cyclic groups ...
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1answer
44 views

Show that $(M\times N)/R\cong M$.

If there are two groups, $M$ (with multiplication $\cdot_M$) and $N$ (with multiplication $\cdot_N$) and we define a new group $M \times N$ with multiplication such that $$ (m,n)(m',n') = (m \cdot_M ...
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0answers
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Functional equations and finding isomorphism between groups

While trying to prove that two groups are isomorphic I noticed how similar the problem is to finding a solution to a functional equation and then proving that the function is bijective. For instance ...
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1answer
55 views

If $\mathbb{Z}_m \times \mathbb{Z}_n \cong \mathbb{Z}_{m'} \times \mathbb{Z}_{n'}$ with $m|n$ and $m'|n'$, then does $m=m'$ and $n=n'$? [closed]

Suppose $\mathbb{Z}_m \times \mathbb{Z}_n $ is isomorphic to $\mathbb{Z}_{m'} \times \mathbb{Z}_{n'}$ as groups, where $m$ divides $n$ and $m'$ divides $n'$. Does that mean $m=m'$ and $n=n'$?
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Classify $\mathbb{Z}_{108}$ up to an isomorphism [closed]

Note: My question is a lot more general. Can you provide me with an answer about what "classify" implies when it comes to groups and what are the steps that one should follow to carry through with ...
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0answers
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Normal groups and homomorphism are the same, and this gives an approach to isomorphism theorem?

I was reading a post here that give some interesting approach about isomorphism theorem (see quote). But there are some things I don't understand. What exactly does this mean? The Second ...
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1answer
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Groups of order 56 with Sylow 2-subgroup isomorphic $Q_8$

I try to classify non-abelian groups of order $56$ with sylow $2$-subgroup isomorphic to quaterion group $Q_8$. More accurately I want to construct $2$ non-isomorphic such groups. This is an excercise ...
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Permutation groups: $S_4$ and $D_4$.

Question. Determine the subgroup of $S_4$ generated by $\sigma=(1\ 2\ 3\ 4)$ e $\tau = (2\ 4)$. Show that $\left<\sigma, \tau\right> <S_4$ is isomorphic to the group of square symmetries. ...
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2answers
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Show that $R^n/Im(\rho)=R^{n-1}\bigoplus R/2R$, where $R$ is an abelian group and $\rho$ is the following function.

Consider the following group homomorphism $\rho$, where $R$ is an abelian group, \begin{align*} \rho:&R\rightarrow R^n\\ \rho(r)=&(2r,2r,\cdots,2r). \end{align*} Show that $R^n/Im(\rho)=R^{n-...
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1answer
58 views

Show that the quotient group $G/N$ contains a subgroup isomorphic to $H$. [closed]

Let $G$ be a finite group, $N\mathrel{\lhd}G$ a normal subgroup of $G$, and $H\leq G$ a subgroup of $G$. Suppose that $|H|$ and $|N|$ are relatively prime (i.e., $\gcd(|H|,|N|)=1$). Show that the ...
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1answer
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If $G$ is a cyclic group of order $d$ with generator $a$, show that $\dfrac{\mathbb{Z}}{d\mathbb{Z}}$ is isomorphic $G$. [duplicate]

We have to $|G|=d$ and $G=\left<a\right>=\{a^n : n \in \mathbb{Z}\}$ by definition of cyclic group. To show what you are asking, I want to define a function, not necessarily this $$\phi : \...
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2answers
42 views

Group Isomorphism regarding Sylow Subgroups

Suppose I have given two groups say, $G_1,G_2$ such that they have same order.I'm assuming they are non commutative.Then their Syllow subgroups has same order clearly.If I'm given that the number of ...
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1answer
36 views

$G$ be a group where $G= \mathbb{Z}_6 \oplus \mathbb{Z}_8$ and the normal subgroup $H=\langle(2,4)\rangle$ use orders of elements to determine…

I have the elements of $H$: $\langle(2,4)\rangle={(2,4),(4,0),(0,4),(2,0),(4,4),(0,0)}$ where $|G/H|=8$ possible isomorphic classes: $\mathbb{Z}_8$ ,$\mathbb{Z}_4\oplus \mathbb{Z}_2$, $\mathbb{Z}_2\...
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2answers
43 views

Finding Homomorphisms from dihedral groups to cyclical groups

Ok so there was another question very similar to this on here however it leaves me a little confused. $\bf{Question}$ Let G = $D_{14}$ the Dihedral group order 14 and A = $c_7$ be the cyclical ...
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1answer
50 views

There are only two groups of order six, up to isomorphism: $\mathbb Z_6$ and $S_3$. [duplicate]

Let $G$ be group with order $6$. Prove that either $G$ and $\Bbb Z_{6}$ are isomorphic binary structure or $G$ and $S_{3}$ are isomorphic binary structure. I know that for isomorphic binary ...
2
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1answer
40 views

Prove $G/(M\cap N) \cong M/(M\cap N) \times N/(M\cap N)$ where $G=MN$ and $M,N\triangleleft G$

If we consider the map $\phi: M\times N \rightarrow M/(M\cap N) \times N/(M\cap N)$, I was able to show that this is onto and the kernel of the map is $(M\cap N) \times (M\cap N)$ and hence by first ...
5
votes
2answers
177 views

Is $S_4 \times C_2$ isomorphic to $(C_2 \times C_2 \times C_2) \rtimes S_3$

Let $S_n$ denote the symmetric group on $n$ letters and $C_n$ denote the cyclic group of order $n$. Consider $(C_2 \times C_2 \times C_2) \rtimes S_3$ where $S_3$ acts on $(g_1, g_2, g_3) \in C_2 \...