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.

61 questions
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11 answers

Amazing isomorphisms

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 they are isomorphic or not?
24
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1 answer

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 $\mathbb{R}^7$ endowed with the Euclidean inner product) such that for…
23
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5 answers

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: $\mathbb{R}/\mathbb{Z} \cong \mathcal{C}$, where $\mathcal{C}$ is the unit circle in the…
19
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7 answers

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, maybe?
Grigory M
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18
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5 answers

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 of modular curves and consider the action on the…
Barry Smith
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16
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3 answers

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 j \pm k}2 \right\} \subseteq…
15
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2 answers

Why isn't $SO(3)\times SO(3)$ isomorphic to $SO(4)$?

Why is $SO(3)\times SO(3)$ isomorphic to $SO(4)$?
15
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2 answers

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 $\mathrm{GL}_2(\mathbb{F}_9)$." (Apparently this has been called the "fake…
14
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3 answers

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 values of the Lie brackets on the canonical bases and…
Sam
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12
votes
2 answers

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 well known that the smallest projective plane,…
9
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1 answer

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) \\ \hline \Spin(2) & \mathrm{SO}(2) \\ \hline \Spin(3)…
whacka
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8
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0 answers

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 binomial theorem can both be reinterpreted as…
8
votes
2 answers

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 ask once again: Why do we have…
7
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2 answers

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

In the tradition of this question, why is $\operatorname{PGL}(2,4)\cong A_5$?
7
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1 answer

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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