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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14 views

Show that the given mapping is isomorphism.

Let's define a mapping from A→Ca by f(Ng)=(g^-1)a(g). where: A=set of right cosets i.e.Ng Ca= Conjugacy Class Can anyone explain that how to show this an isomorphism please?
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Show that he function mapping is isomorphism. [closed]

show that the given mapping is Isomorphism by explaining all steps in details i e. Well Defined, One-One, Onto
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About isomorphism of two groups. [closed]

If two groups have same number of subgroups of same order , then do they need to be always isomorphic? Ok, that is question which arrived in my mind , when I was going to prove that there is an unique ...
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1answer
41 views

Isomorphism between $\langle c,d:c^n,d^2,(cd)^2\rangle$ and $\langle a,b:a^2,b^2,(ab)^n\rangle.$

Let $C :=\langle c,d : c^n , d^2, (cd)^2\rangle, A := \langle a,b : a^2 , b^2, (ab)^n\rangle.$ Show that the map $\phi: C\mapsto A: \phi(c) = ab,\phi(d) = b$ is an isomorphism, where $\langle S : R\...
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1answer
33 views

A group isomorphism involving dihedral group

Show that $$\begin{align} \langle c, d: c^n , d^2, (cd)^2\rangle &\to D_n,\\ c&\mapsto r,\\ d&\mapsto s\end{align}$$ is an isomorphism, where $D_n$ is the dihedral group, $r$ represents ...
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Let $G$ be a group of order $n$ and $H$ a subgroup of order $k$ [closed]

Let $G$ be a group of order $n$ and $H$ a subgroup of order $k$. If $H$ is a normal subgroup, then prove that $G$ is isomorphic to a subgroup of $S_{n/k}$.
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4answers
71 views

Every non abelian group of order 6 is isomorphic to $\displaystyle S_{3}$ (the symmetry group of order 6).

Every non abelian group of order $6$ is isomorphic to $\displaystyle S_{3}$ (the symmetry group of order $6$). I tried to prove it but got stuck mid-way. Let $\displaystyle G$ be a non-abelian group ...
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1answer
62 views

How to prove $H_1(\mathbb{C}/\Lambda, \mathbb{Z})$ to the lattice $\Lambda$ ,$\gamma \mapsto \int_\gamma dz$ is well defined?

Let fix an arbitraly lattice of $\mathbb{C}$ and call it Λ. How to prove $H_1(\mathbb{C}/\Lambda, \mathbb{Z})$ is isomorphic to the lattice $\Lambda$ via the map $\gamma \mapsto \int_\gamma dz$ ? If ...
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Outer automorphisms of extraspecial groups in GAP

Let $G_n$ be the extraspecial group of order $2^{1+2n}$. Its outer automorphism group is known to be isomorphic to the general orthogonal group $GO(2n)$. I'd like to get an explicit map of this ...
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163 views

If $\displaystyle \bigoplus_{i=1}^{n} \mathbb{Z} \cong \bigoplus_{i=1}^{m} \mathbb{Z}$ as groups, then $n=m.$

If $\displaystyle \bigoplus_{i=1}^{n} \mathbb{Z} \cong \bigoplus_{i=1}^{m} \mathbb{Z}$ as groups, then $n=m.$ Here is my proof: Let $G$ be a group such that $\varphi:G \rightarrow \displaystyle \...
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3answers
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Show that $G_1$ and $G_3$ are not isomorphic

While practising for an exam I encountered the following question. Consider the groups $G_1=(\mathbb{Z}/100\mathbb{Z})^{\times}$ and $G_2=(\mathbb{Z}/110\mathbb{Z})^{\times}$ and $G_3=(\mathbb{Z}/132\...
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2answers
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Are the groups $(\mathbb{Z}/n\mathbb{Z})^{\times}$ and $(\mathbb{Z}/m\mathbb{Z})^{\times}$ isomorphic if they have the same order?

Let $(\mathbb{Z}/n\mathbb{Z})^{\times}$ the set of units of $\mathbb{Z}/n\mathbb{Z}$ (i.e the elements with multiplicative inverse). And consider the groups: $$(\mathbb{Z}/17\mathbb{Z})^{\times}, (\...
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Maximum number of elements (excluding the identity)that one need to compute the order, to determine the isomorphism class of Abelian group $G$

I just finished below to exercises. If $G$ is Abelian group of order $9$ then, maximum number of elements (excluding the identity)that one need to compute the order, to determine the isomorphism ...
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1answer
31 views

Is $\langle S \rangle / \operatorname{nc}(T) \simeq \langle S \setminus T \rangle$, where $T \subseteq S$?

Let $S$ be a set and $\langle S \rangle$ denote the free group generated by the set $S$. If we take a subset $T$ of $S$ and consider the quotient group $\langle S \rangle / \operatorname{nc}(T)$, ...
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1answer
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Understanding an isomorphism between direct limit of the character group and character group of the inverse limit

I am struggling with the following general setup from Chapter IV(page 269) in Macdonald's book on symmetric functions and Hall polynomials. Let $$K=\varprojlim M_n$$ be their inverse limit, which is ...
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1answer
57 views

Proving that the group $U(30)$ is isomorphic to the group $\Bbb Z_8$.

I am trying to prove that the group $U(30)$ is isomorphic to the group $\Bbb Z_8$. I know that both groups have an order of $8$ and are both cyclic, but this isn't enough to show that an isomorphism ...
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16 views

Example of a pair of non-isomorphic quasi-groups with parastrophic Latin squares?

A Latin square $\Lambda$ over an alphabet $A$ is a set of triples of elements of $A$ such that for every $\alpha,\beta\in A$, there is exactly one $\gamma\in A$ for which $(\alpha,\beta,\gamma)\in \...
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1answer
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Let $G$ be abelian of order $p_1^{m_1}p_2^{m_2}…p_k^{m_k}$. Show $G\cong P_1\times …\times P_k$ where $P_i$ are subgroups of order $p_i^{m_i}$

I assume I should prove that $G$ is equal to inner product $P_1P_2\cdots P_k$ and that there would be natural isomorphism. Since $G$ is abelian all its subgroups are normal but I don't know why ...
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1answer
45 views

Help with isomorphic groups?

I have been asked to give all the non-isomorphic, abelian groups with order $12$ & order $15$ respectively. The answer reads Order $12$: $\Bbb Z_{12}$ and $\Bbb Z_6 \times\Bbb Z_2$ Order $15$: $\...
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36 views

Number of binary operations isomorphism with 4 elements

Let $A = \{1, 2, 3, 4\}$. What is the number of binary operations $*$ defined on $A$, such that $(A,*)$ is a group isomorphism to $(\Bbb Z_4,+)$, and the order of $3$ is $2$ in this group ? I tried ...
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1answer
61 views

Prove $\operatorname{Gal}(EL/E)$ is a Galois extension in this case

I've been solving problems from my Galois Theory course, and I don't know how to approach this one. It says: Given $K/F$ field extension, and let $E,L$ be in-between fields ($F\subseteq E\subseteq K$...
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Calculating $\Bbb R_+^\Bbb C:={\mathbb R_+} \otimes_{\mathbb R_+} {\mathbb C}$

I would like to take the vector space $\Bbb R_+$ via transport map $\exp:\Bbb R\to \Bbb R_+$ and make it into a complex vector space over a complex field. So I need to calculate the complexification: $...
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32 views

Are groups isomorphic to direct product of normal subgroups?

Consider a group $G$ such that $|G|=35=5*7$. It has Sylow $5$-groups $P$ of order $5$, hence $P\simeq\mathbb Z_5$, and Sylow $7$-groups $Q$ of order $7$, hence $Q\simeq \mathbb Z_7$. The Sylow ...
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1answer
29 views

Lie group isomorphism between U(2)/U(1) and SU(2)

I know that $SU(2)$ is diffeomorphic to 3-sphere $S^3$. And, I know that $U(2)/U(1)$ is diffeomorphic to $S^3$. I would like to know if there is an Lie group isomorphism between $U(2)/U(1)$ and $SU(2)$...
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1answer
22 views

Does a topological isomorphism between monothetic groups preserve the generator?

Suppose that $\phi$ is a topological isomorphism between the topological groups $G=\overline{\{x^{n}:n\in\mathbb{Z}\}}$ and $H=\overline{\{y^{n}:n\in\mathbb{Z}\}}$. Can we conclude that $\phi(x)=y$? ...
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1answer
62 views

Why the following implies that $G \cong H$?

If I have the following sequence: $$J \xrightarrow{v} H \xrightarrow{u} G \rightarrow 0 \qquad (1)$$ Where $v$ is the zero map. Does that mean $G \cong H$ i.e., $G$ isomorphic to $H$?if so why? Or, do ...
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1answer
69 views

$SL(2,\mathbb{C})\cong \operatorname{Spin}(1,3,\mathbb{R})?$ or $SL(2,\mathbb{C})\cong \operatorname{Spin}(3,1,\mathbb{R})?$ [closed]

Which one of the isomorphism is correct? $$SL(2,\mathbb{C})\cong \operatorname{Spin}(1,3,\mathbb{R})?$$ $$SL(2,\mathbb{C})\cong \operatorname{Spin}(3,1,\mathbb{R})?$$ If $SL(2,\mathbb{C})\cong \...
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32 views

Finding number of surjective homomorphisms from $S_4$ to various groups without the First Isomorphism Theorem

I am working through a problem where I have shown that a group homomorphism $\phi :G \longrightarrow H$ is uniquely determined by the image of its generators, and also by $\text{ker}\;\phi$ and the ...
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2answers
82 views

Group action on topological spaces.

Let $X$ be a simply connected topological space, and $G$ a group that acts faithfully on $X$ by homeomorphism (i.e. the map $x \mapsto g \cdot x$ is a homeomorphism for each $g \in G)$. Let $X / G$ be ...
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1answer
46 views

Rationals with denominator $1$ or $2$ under addition

Consider $G := \{ a/b : \gcd(a, b) = 1 \text{ and } |b| \leq 2 \} \subset \mathbb{Q}$. This is not a subgroup of $\mathbb{Q}^\times$ since $\frac 1 2 \cdot \frac 3 2= \frac 3 4$. However, addition in $...
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1answer
54 views

Isomorphism: minimal number of steps to check

There are in general two methods to check a map is a isomorphism. The first is to show it is injective, surjective and a homomorphism. A second way is via the construction of inverse. I got kind of ...
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1answer
51 views

Show that $\prod_{i=1}^{n}\text{Aut}(G_i)\cong \text{Aut}\Big(\prod_{i=1}^{n}G_i\Big)$ (intuition on smaller $n$)

Let $G_1,...,G_n$ be groups of finite order such that theirs orders $|G_i|$ are pairwise coprime. Show that $\xi:$$\prod_{i=1}^{n}\text{Aut}(G_i)\cong \text{Aut}\Big(\prod_{i=1}^{n}G_i\Big)$. Here I ...
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1answer
29 views

Group isomorphism and the structural similarity

A question on group isomorphism. Let (A,・) and (B,*) be some groups, and I want to show they're isomorphic. I know there has to exist a bijection function f: Α → Β such that f(x・y)=f(x)*f(y) for all x,...
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2answers
98 views

automorphism, endomorphism, isomorphism, homomorphism within $\mathbb{Z}$

From Wikipedia: An invertible endomorphism of $X$ is called an automorphism. The set of all automorphisms is a subset of $\mathrm{End}(X)$ with a group structure, called the automorphism group of $X$ ...
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1answer
58 views

irreducible representations of $\mathbb{Z}_n$

I'm trying to find all the irreducible representations (which are not isomorphic) of $(\mathbb{Z}_n,+)$ in the field of real numbers. I saw that we can realize by mapping a group generator $g$ to $1$ ...
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3answers
103 views

Why is $C_4 \times C_4$ not isomorphic to $C_4 \times C_2 \times C_2$?

a while ago I was trying to prove this: Show that $C_4 \times C_4$ is not isomorphic to $C_4 \times C_2 \times C_2$. I know that we can write $C_4 \times C_2 \times C_2$ as $C_4 \times V_4$, where $...
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1answer
50 views

What is the relationship between $S0(2)$ and $PSL(2,R)$?

The Holonomy of a hyperbolic surface S in terms of differential geometry is either $SO(2)$ or $O(2)$ depending on Orientability. And a hyperbolic structure as a special (X,G)-structure: $\pi_1(S)⊂PSL(...
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1answer
45 views

Show that $\phi : \mathbb{Z}_2\rightarrow\mathbb{Z}_4^*$ is an isomorphism.

I know that $\mathbb{Z}_2=\{0,1\}$ and $\mathbb{Z}_4^*=\{1,3\}$ So $\phi$ is defined as:$\phi(0)=1$ and $\phi(1)=3$ Clearly then it is bijective since every element of $\mathbb{Z}_2$ maps to exactly ...
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2answers
109 views

Let $G$ and $H$ be nontrivial groups such that $G$ is simple and let $f : G \to H$ be a surjective homomorphism. Show that $f$ is an isomorphism. [closed]

Having some discomfort in my solution and was wondering if there was an easier way to do this, Thanks. If $G$ is simple, then the kernel of $f$ is either $\{1\}$ or $G$ itself. Since the kernel is a ...
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0answers
59 views

Second isomorphism theorem proof for groups.

I'd like to be prove the second isomorphism theorem. Let $H$ and $K$ be two subgroups of $G$ with $K$ normal in $G$. The subgroup $H \cap K$ is normal in $H$, $HK$ is a subgroup of $G$, and $K$ is ...
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1answer
62 views

A group isomorphism between $\mathbb{Q/Z}$ and $\mathbb{Q/2Z}$

Question: Prove that $\mathbb{(Q/Z, +)}\cong\mathbb{(Q/2Z, +)}$ My attempt To prove they are isomorphic I need to define a map from $\mathbb{Q/Z}$ to $\mathbb{Q/2Z}$ which is bijective and preserve ...
2
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1answer
67 views

counter example needed: existence of an isomorphism which maps a “diagonal” (non product) subgroup of a finite abelian group to a product subgroup.

Consider a finite abelian group: $$G \cong \mathbb{Z}/p_1^{\alpha_1}\mathbb{Z} \times\mathbb{Z}/p_2^{\alpha_2}\mathbb{Z} \times\dots\times\mathbb{Z}/p_n^{\alpha_n}\mathbb{Z} $$ Let $K$ be a subgroup ...
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0answers
17 views

Twisted subgroups. (Subgroups of $D_5 \times S^1$

Heya all I am looking for twisted subgroups of $D_5$, equivalently subgroups of $D_5 \times S^1$, and was wondering if the group generated by the elements $[\rho,0]$ and $[\kappa,\frac{1}{2}]$ forms ...
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3answers
25 views

If $G$ is a group, $H \leq G$, and for $a \in G$, $x \mapsto axa^{-1}$ is an isomorphism. Why isn't this sufficient to conclude $H$ is normal? [closed]

A subgroup $H$ is normal if and only if $\forall a \in G (H = aHa^{-1})$, $a H a^{-1} \subseteq H$ and for arbitrary $a \in G$, $x \mapsto axa^{-1}$ is an isomorphism, why doesn't this imply that $H$ ...
0
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0answers
25 views

Role of semidirect product and intuition for $O(2)\simeq U(1)\rtimes \mathbb{Z}_2$?

I have a very basic understanding of some common groups, and I'm trying to get some intuition for this isomorphism. My thinking so far is that $O(2)$ is rotations and reflections in $\mathbb{R}_2$, ...
0
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1answer
62 views

Are these two infinite groups isomorphic? [duplicate]

Question. Prove or disprove that $G_1$ and $G_2$ are isomorphic. $$G_1=\mathbb Z_5 \times \mathbb Z_{5^2}\times \mathbb Z_{5^3}\times \mathbb Z_{5^4} \times \cdots$$ $$G_2= \mathbb Z_{5^2}\times \...
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0answers
43 views

Cartesian Product of Quotient groups

Let $A, B$ be groups and $C,D$ be normal subgroups of $A,B$ respectively. The question was proving $A/C\times B/D\cong (A\times B)/(C\times D)$; which can be solved with 1st homomorphism theorem. ...
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0answers
67 views

Theorem of homomorphism

I have one question about group theory. Our professor told us that there is a standard type of problems when it comes to proving that two groups are isomorphic. And he was always using the First ...
3
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0answers
64 views

Equivalence of two elements of the free group under automorphisms?

I have two elements of the form $$ w = x_1^{a_1} x_2^{a_2} x_3^{a_3} x_4^{a_4} x_5^{a_5} x_6^{a_6} $$ and $$ w' = x_1^{b_1} x_2^{b_2} x_3^{b_3} x_4^{b_4} x_5^{b_5} x_6^{b_6} $$ for integers $a_i$ and $...
2
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1answer
75 views

About a collection of linear maps $L=\bigcup_{i=1}^\infty X_i$

I recently asked: About this transformation, and I wanted to follow up with another related question: Consider a collection of linear maps $L=\bigcup_{i=1}^\infty X_i.$ Where $X_1=\big\{(x,y)\mapsto\...

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