# 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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### 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 ...
0answers
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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 $U$. ...
2answers
270 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 ...
2answers
583 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 ...
1answer
71 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 ...
1answer
273 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 $2n-1$) ...
0answers
73 views

### 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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### 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 ...
2answers
417 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 $\mathrm{GL}_2(\mathbb{F}_9)$...
1answer
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### 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 ...
1answer
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### 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 ...
5answers
894 views