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Questions tagged [exceptional-isomorphisms]

Isomorphisms between members of infinite families of mathematical objects (finite simple groups, Lie groups etc), that are not examples of a pattern of such isomorphisms.

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Adjoint functors, inclusion functor, reflective subcategory

Suppose that a category $A$ is reflective in a category $B$ and that the inclusion functor $K:A\to B$ has a left adjoint $F:B\to A.$ Now what does it technically mean that this bijection of sets $$A(...
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0answers
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Could every finite simple group be related to a pair of Lie Groups?

In terms of the classification of simple groups, it is known that every finite simple group is either: Cyclic Alternating Of Lie type One of the 26 Sporadic groups On the other hand there are 5 ...
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0answers
43 views

Signature interpretable in a category K

Here on the page 10, there is a notion of a finitary relation symbol interpretable in a category $\cal K$ with $U:{\cal K}\to \mathbb {Set}$ whose morphisms are monomorphisms preserved by $U$. ...
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2answers
93 views

To complete the proof that $\operatorname{PSL}(2,\Bbb F_5)\cong A_5$

Elementary isomorphism between $\operatorname{PSL}(2,5)$ and $A_5$ I want to fill in the details of user David Speyer's algebraic proof that $\operatorname{PSL}(2,\Bbb F_5)\cong A_5$. I haven't ...
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2answers
228 views

Proof of isomorphism between $\text{PGL}_2(\mathbb{F}_5)$ and $S_5$

This question has been asked here before but I don't think any of the previous answers are clear to someone like me who only has an elementary background in abstract algebra. So can I take the time to ...
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1answer
45 views

Find if $Z[X]/<x^2-27> $ is isomorphic or not with $Z[\sqrt3]$

$Z[\sqrt3$] = $\{a + b\sqrt3 \mid a,b \in Z \}$ To find out if the statement is true or not i tried the following. I created a surjective morphism that has $<x^2-27>$ as a kernel. And this way ...
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1answer
176 views

The duad-syntheme-total construction for higher values of $n$

Let $X=\{1,\cdots,2n\}$. Call a $2$-subset of $X$ a duad. Let $D$ be the collection of all $X$'s duads. Call a partition of $X$ into duads a syntheme. Call a partition of $D$ into (necessarily $2n-1$) ...
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0answers
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Tetrality and ${\frak so}(8)$'s three irreps

Over at the $n$-Category Cafe John Baez talks about something called "tetrality." This is where $S_4$ acts on the complex lie algebra $\mathfrak{so}(8,\mathbb{C})$ in a way that factors through the ...
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0answers
102 views

Why $\mathrm{SU}_2(\mathbb{C})/\lbrace \pm 1 \rbrace \cong \mathrm{SO}_3(\mathbb{R})$ is exceptionnal?

One can prove that $\mathrm{SU}_2(\mathbb{C})/\lbrace \pm 1 \rbrace \cong \mathrm{SO}_3(\mathbb{R})$. But why we call this isomorphism exeptionnal? I believe that we call it exptionnal because there ...
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1answer
25 views

Questions on Isomorphisims

$\mathbb{N}$, the set of natural numbers is set theoretically isomorphic to $\mathbb{Z}$,the set of integer. But my questions is why it is isomorphic to $\mathbb{Q}$, the set of rational numbers but ...
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1answer
288 views

Spin(4,1) = Sp(1,1) isomorphism

I am interested in the exceptional isomorphism Spin(4,1) = Sp(1,1). The correspondance is already mentioned here: spin group Spin(4,1) but the explicit isomorphism is not given. I would like to know ...
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5answers
147 views

Seeing $PSL_2(\mathbb{C}) \cong SO_3(\mathbb{C})$

How can I see the isomorphism between projective special linear group (order 2) and the special orthogonal group (order 3)? I know only this setting $PSL_2(\mathbb{C}) = SL_2(\mathbb{C})/Z(SL_2(\...
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11answers
2k views

Amazing isomorphisms [closed]

Just as a recreational topic, what group/ring/other algebraic structure isomorphisms you know that seem unusual, or downright unintuitive? Are there such structures which we don't yet know whether ...
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0answers
39 views

How can I prove that $PSL(2,3)\simeq A_4$?

I would like to prove that $PSL(2,3) \simeq A_4$. I know the structure of $SL(2,3)$ and that $A_4=V\rtimes C_3$, where $V$ is the Klein subgroup. How can I proceed? Thanks for the help!
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0answers
42 views

Find an isomorphism $PGL_2(F_3) \cong S_4$

I am struggling to find an explanation why this true, I know and I'm sorry that kind of question is commonly asked , although I couldn't find anything about this particular question. Help is ...
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2answers
192 views

Projectivization and stereographic projection, or: Why nonlinear + nonlinear = linear?

This previous question had me thinking about something I've taken for granted. Consider $\mathrm{SL}_2(\mathbb{C})$ acting on $\mathbb{C}^2$, which descends to an action on the complex projective line ...
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1answer
229 views

The last accidental spin groups

For dimensions $n\le 6$ there are accidental isomorphisms of spin groups with other Lie groups: $\DeclareMathOperator{Spin}{\mathrm{Spin}}$ $$\begin{array}{|l|l|} \hline \Spin(1) & \mathrm{O}(1) \...
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0answers
112 views

Is SU(2) a subgroup of the exceptional lie group $G_2$?

I am not an expert in Lie groups and I have spent ages looking at textbooks; I assume that because I haven't found this statement explicitly it must either be untrue or obvious ;) The only thing I ...
2
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1answer
126 views

$SL_2(\mathbb Z_3)/Z(SL_2(\mathbb Z_3)) \cong \mathbb A_4$

I am trying to prove that the quotient group $SL_2(\mathbb Z_3)/Z(SL_2(\mathbb Z_3))$ is isomorphic to $\mathbb A_4$. I could show that $SL_2(\mathbb Z_3)$ has $24$ elements (one can see this ...
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1answer
140 views

Tensor products isomorphic to hom-sets with a structure

In which cases the tensor product of objects, say A and B, is (isomorphic with) the objects with the carrier set Hom(A,B) and a corresponding structure?
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1answer
67 views

Isomorphism between $E$ and $Lin(E)$ : infinite dimensional case.

Does $E$ and $Lin(E)$ where $Lin(E)=\{A : E \rightarrow E ∣ A \quad\text{is linear}\}$ are isomorph if $E$ infinite dimensional case ? I know that if $E$ is finite dimension the result is true if ...
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1answer
567 views

Discrete math Group - Isomorphism and Automorphism

Let G be a Cyclic group Prove or disprove: A.let $ a,b \in G \quad $ so the function $ f:G \to G,f(a^k) = b^k$ is Automorphism of G(which means G is Isomorphism to herself) B.let a,b generators ...
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2answers
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isomorphic quotient rings?

I have trouble in determining, whether two rings are isomorphic: Let's have $R = GF(3)$ and rings $R[x]/(x^2+x+2)$ and $R[x]/(x^2+2x+2)$. How can one determine whether these two rings are isomorphic?...
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1answer
28 views

Are there infinite positive $n \in \Bbb N$ such that ($D^+_n$, |) is isomorphic to ($D^+_{4100}$, |)?

Are there infinite positive $n \in \Bbb N$ such that ($D^+_n$, |) is isomorphic to ($D^+_{4100}$, |)? I'm guessing no because I can't relate every element of ($D^+_{4100}$, |) to ($D^+_n$, |) because ...
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1answer
75 views

How can I prove that $(ℤ, ≺)$ is not isomorphic to $(ℕ, ≤)$

We define the relation $≺$ between pairs of integers like this: $n≺m$ is true if and only if one of the following conditions holds: a) $0≤n≤m$ b) $0≤n$ and $m<0$ c) $n<0 , m<0$ and $|n|≤...
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1answer
162 views

Abstract Algebra: Find a number of non-isomorphic subgroup of $S_3$

I'm having difficulty in understanding the method to find the solution for this question. I repeat Question: Find the number of non-isomorphic subgroup of $S_3$ So is this the way to find the ...
3
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2answers
577 views

Prove that $PSL(2,9)\cong A_6.$

I have tried to solve the Exercise 8.12, page 227, in Joseph J. Rotman, An Introduction to the Theory of Groups, Fourth Edition, Graduate texts in Mathematics, Springer-Verlag, New York (1995). The ...
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1answer
286 views

Question regarding isomorphisms in low rank Lie algebras

I am reading Brian Hall's book 'Lie Groups, Lie Algebras, & Representations' and on p.271 I find that in low rank Lie algberas there are some isomorphisms. For example, $\mathfrak{sl}(2;\mathbb{C})...
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5answers
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Examples of non-obvious isomorphisms following from the first isomorphism theorem

I am learning the first isomorphism theorem, and I am working with some isomorphisms to practice for my upcoming test. I know some of the basic ones like: $\mathbb{R}/\mathbb{Z} \cong \mathcal{C}$, ...
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1answer
664 views

Why is $PGL_2(5)\cong S_5$?

Why is $PGL_2(5)\cong S_5$? And is there a set of 5 elements on which $PGL_2(5)$ acts?
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3answers
792 views

Binary tetrahedral group and $\rm{SL}_2(\mathbb F_3)$

The binary tetrahedral group $\mathbb T$ is an interesting 24-element group. For instance it can be expressed as the subgroup $$ \mathbb T = \left\{ \pm 1, \pm i, \pm j, \pm k, \dfrac{\pm 1 \pm i \pm ...
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2answers
277 views

How to count the conjugates of an exotic $S_5$?

It can be read off the The Elliott configuration - a $5$-coloring of $K6$ - that $S_6$ has an exotic $S_5$ subgroup (it's not a point stabilizer) which I will call $X_5 = \langle (1\;3\;6\;4\;5), (1\;...
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1answer
134 views

Give an isomorphism between the linear spaces $U\times V$ and $L(U,V)$

I was hoping someone could help me with the above question (give an isomorphism between the linear spaces $U\times V$ and $L(U,V)$. I have a hunch that I should work with the bases of U and V but the ...
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1answer
126 views

In $D_4$, find $H_1$ isomorphic to $\mathbb{Z}_4$ and $H_2$ isomorphic to the Klein Four Group with $D_4/H_1$ isomorphic to $D_4/H_2$.

a) In $D_4$, find $H_1$ isomorphic to $\mathbb{Z}_4$ and $H_2$ isomorphic to $V$, the Klein Four group, with $D_4/H_1$ isomorphic to $D_4/H_2$. b) In $D_4$, find subgroups $H$ and $K$ with $H$ normal ...
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3answers
196 views

Why are these elements generators of cyclic groups?

My example says: AutC$_4$, $C_4 = \{1, a, a^2, a^3\}$ And then it points to $a$ and $a^3$ and says these are generators and these are the two elements of order 4. Why is this?
2
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1answer
238 views

Why is $\operatorname{Spin}(3) \cong \operatorname{SU}(2)$?

Why is $\operatorname{Spin}(3) \cong \operatorname{SU}(2)$? I can't seem to realize $\operatorname{Spin}(3)$ explicitly as a matrix group and so I'm having issues constructing an isomorphism, of ...
16
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5answers
2k views

Elementary isomorphism between $\operatorname{PSL}(2,5)$ and $A_5$

At this Wikipedia page it is claimed that to construct an isomorphism between $\operatorname{PSL}(2,5)$ and $A_5$, "one needs to consider" $\operatorname{PSL}(2,5)$ as a Galois group of a Galois cover ...
10
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2answers
201 views

Isomorphism between $\mathfrak o(4,\mathbb R)$ and $\mathfrak o (3,\mathbb R) \oplus\mathfrak o (3,\mathbb R) $

I've been trying to find a Lie algebra isomorphism $$\mathfrak o(4,\mathbb R)\cong\mathfrak o (3,\mathbb R) \oplus\mathfrak o (3,\mathbb R) $$ but haven't managed so far. I have written down the ...
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2answers
820 views

Why is $PGL(2,4)$ isomorphic to $A_5$

In the tradition of this question, why is $\operatorname{PGL}(2,4)\cong A_5$?
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2answers
3k views

Why is $SO(3)\times SO(3)$ isomorphic to $SO(4)$?

Could you please explain me the reason why they are isomorphic? Thanks, bye!
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7answers
2k views

Why $PSL_3(\mathbb F_2)\cong PSL_2(\mathbb F_7)$?

Why are groups $PSL_3(\mathbb{F}_2)$ and $PSL_2(\mathbb{F}_7)$ isomorphic? Update. There is a group-theoretic proof (see answer). But is there any geometric proof? Or some proof using octonions, ...