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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Are all isomorphisms between the Fano plane and its dual of order two?

Famously from every finite projective plane $P$ we can create a dual projective plane $P'$ by taking the points of $P'$ to be the lines of $P$ and the lines of $P'$ to be the points of $P$. It is also ...
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What is $(\operatorname{SL}_2(\mathbb{R})\times\operatorname{SL}_2(\mathbb{R})\times\operatorname{SL}_2(\mathbb{R}))/\{\pm(I_2,I_2,I_2)\}$?

I know that $\operatorname{SL}_2(\mathbb{R})/\{\pm\operatorname{I}_2\}=\operatorname{PSL}_2(\mathbb{R})$ and that $(\operatorname{SL}_2(\mathbb{R})\times\operatorname{SL}_2(\mathbb{R}))/\{\pm(\...
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Permutations of a $4\times4$ grid generated by permutations of rows, columns, and $2\times2$ quadrants

Over here it is claimed the group $G$ of permutations of a $4\times4$ grid generated by the subgroups $R$ of row permutations, $C$ of column permutations and $B$ of block permutations (of the four $2\...
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3 votes
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transitive action from $A_5$ on a set of $6$ elements

Can we find a transitive action from the alternating group $A_5$ on the set $X$ with 6 elements? I think we can't because the $|X|=6$ which greater than $5$. It also because $A_4$ does not have a 6-...
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4 votes
2 answers
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The Homogeneous Space $SO^+(1,3)/ \text{Sim}(2)$

I'm trying to understand the homogeneous space $SO^+(1,3)/\text{Sim}(2)$. In wikipedia it is said that "$SO^+(1,3)/\text{Sim} (2)$ is the Kleinian geometry that represents conformal geometry on ...
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2 votes
1 answer
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Dimensional coincidences between Lie groups

There are several exceptional isomorphisms between classical Lie groups that occur in low dimensions; of course it is necessary for the dimensions of the groups to coincide for this to happen! I'm ...
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5 votes
1 answer
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Implications of $\text{Spin}(3)\times \text{Spin}(3) \cong \text{Spin}(4)$

I recently learned that $\text{Spin}(3)\times \text{Spin}(3) \cong \text{Spin}(4)$, and find it intriguing: it seems like the sort of thing that would have lots of interesting consequences. What are ...
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8 votes
0 answers
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Categorifying $1^2+2^2+3^2+\cdots+24^2=70^2$

Does $1^2+2^2+3^2+\cdots+24^2=70^2$ or a simple equivalent have an interesting categorification? Lots of combinatorical identities do. For instance, the Vandermonde convolution identity and the ...
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4 votes
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Extending the action $S_5$ on $2$-subsets of $\{1,\cdots,5\}$ to an action of $S_6$.

The symmetric group $S_5$ acts on the set $\binom{5}{2}$ of ten $2$-subsets of $[5]=\{1,\cdots,5\}$. In The Finite Simple Groups (Wilson), problem 2.21 asks the reader to extend the group action $S_5\...
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Different Lie groups of the same exceptional Lie algebra? for $\mathfrak{g_2,f_4,e_6,e_7,e_8}$

An exceptional Lie algebra is a complex simple Lie algebra whose Dynkin diagram is of exceptional (nonclassical) type. There are exactly five of them: $\mathfrak{g}_{2}$, ${\mathfrak {f}}_{4}$,${\...
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2 votes
1 answer
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About Fano plane symmetries

I have been looking for realizations of order $21$ metacyclic group. I asked about this yesterday and learned some very good information about it Realization of the metacyclic group of order 21 I was ...
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4 votes
0 answers
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How is $\mathbb{F}_2^4$ related to an $8$ element set?

I am trying to understand the part of this answer explaining why $A_8\cong\mathrm{PSL}_4(\mathbb{F}_2)$. Let $|X|=8$. We can form the free vector space $\mathbb{F}_2X=\mathbb{F}_2^8$ with the usual ...
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2 votes
1 answer
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Quotient of binary icosahedral group by its center, i.e., $2I/\{\pm 1\}$ is isomorphic to $A_5$

I know that the inner automorphisms of the binary icosahedral group is given by the quotient $2I/\{\pm 1\}$ where $\{\pm 1\}$ is the center of $2I$. I also know the elements of the binary icosahedral ...
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is it a stronger property of equalizers?

I understand that if $h$ is an eqalizer of $f,f:A\rightrightarrows B$ then $h$ is an isomorphism. Is it stronger to say that if an isomorphism $h$ is an equalizer of $f,f:A\rightrightarrows B$ then $...
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Which of these "minimal" semisimple Lie algebras aren't simple?

The semisimple Lie algebras, indexed by their Dynkin diagrams, are classified as direct sums of the algebras $$ \mathfrak{sl}_{n+1} \quad \mathfrak{so}_{2n+1} \quad \mathfrak{sp}_{2n} \quad \mathfrak{...
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0 votes
4 answers
163 views

mathamatical structure where a bijection is not an isomorphism

During a lecture in module theory, my lecturer mentioned that not in all mathematical structures have the property where a function is an isomorphism if and only if it is a bijection, although this is ...
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24 votes
1 answer
542 views

Is the seven-dimensional cross product unique?

I'm confused about how many different 7D cross products there are. I'm defining a 7D cross product to be any bilinear map $V \times V \to V$ (where $V$ is the inner product space $\mathbb{R}^7$ ...
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3 votes
1 answer
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Show that $\mathfrak{so}(4)\cong \mathfrak{so}(3)\oplus \mathfrak{so}(3)$ [duplicate]

I want to show that $\mathfrak{so}(4)\cong \mathfrak{so}(3)\oplus \mathfrak{so}(3)$. I know that as lie groups $SO(4)\cong (SU(2)\times SU(2))/\mathbb{Z}_2$ and that as $SU(2)/\mathbb{Z}_2 \cong SO(3)$...
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  • 335
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1 answer
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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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3 votes
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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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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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  • 3,578
1 vote
2 answers
328 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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8 votes
2 answers
747 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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0 votes
1 answer
79 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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4 votes
1 answer
291 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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2 votes
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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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2 votes
0 answers
112 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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15 votes
2 answers
479 views

The "fake $\mathrm{GL}_2(\mathbb{F}_3)$" and the binary octahedral group

In this answer, it is mentioned that the binary octahedral group can be realized as $\mathrm{GL}_2(\mathbb{F}_3)$, with "certain elements replaced with scalar multiples in $\mathrm{GL}_2(\mathbb{F}_9)$...
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  • 3,189
0 votes
1 answer
29 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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  • 355
4 votes
1 answer
570 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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  • 43
4 votes
5 answers
1k 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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  • 600
46 votes
11 answers
3k 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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1 vote
0 answers
53 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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2 votes
0 answers
115 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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  • 21
2 votes
2 answers
405 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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9 votes
1 answer
439 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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0 votes
1 answer
183 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 ...
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3 votes
1 answer
344 views

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

I am trying to prove that the quotient group $SL_2(\mathbb Z_3)/Z(SL_2(\mathbb Z_3))$ is isomorphic to $A_4$. I could show that $SL_2(\mathbb Z_3)$ has $24$ elements (one can see this calculating all ...
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  • 3,589
4 votes
2 answers
283 views

Group homomorphism from $SL_2 (\mathbb Z / 5 \mathbb Z)$ to $S_5$

I need to find a group homomorphism from $SL_2 (\mathbb Z / 5 \mathbb Z)$ to $S_5$. There is obviously the trivial homomorphism but I am then asked to deduce that $SL_2 (\mathbb Z / 5 \mathbb Z)/\{I,-...
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0 votes
1 answer
298 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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  • 1
2 votes
1 answer
83 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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  • 1,556
0 votes
1 answer
705 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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  • 65
-1 votes
2 answers
309 views

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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  • 91
3 votes
1 answer
32 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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0 votes
1 answer
105 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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  • 107
0 votes
1 answer
191 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 ...
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3 votes
2 answers
945 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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  • 227
3 votes
1 answer
531 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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  • 4,960
23 votes
5 answers
4k views

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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7 votes
1 answer
1k 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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