# 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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### 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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### 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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### 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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### 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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### 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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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 $S_5\... • 14.6k 1 vote 0 answers 68 views ### Different Lie groups of the same exceptional Lie algebra? for$\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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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### $SL_2(\mathbb Z_3)/Z(SL_2(\mathbb Z_3)) \cong A_4$

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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### 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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### 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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