Articles of exceptional isomorphisms

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, $\mathfrak{sl}(2;\mathbb{C})$ is isomorphic to $\mathfrak{so}(3;\mathbb{C})$. There is a nice way to understand this, which is quite standard (just the complexified Lie algbera version of the connection […]

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 complex plane, under the isomorphism $$x+\mathbb{Z}\mapsto e^{2\pi x i}$$ $\dfrac{\mathbb Z \times \mathbb Z}{\langle (m,n)\rangle}\cong\mathbb Z$, […]

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 $\mathbb{CP}^1$, which we may think of as the Riemann sphere $\widehat{\mathbb{C}}=\mathbb{C}\cup\{\infty\}$. It acts by Möbius transformations. By stereographic projection, $\widehat{\mathbb{C}}$ may be identified with a literal sphere $\mathbb{S}^2\subset […]

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?

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 twelve ramified points. While this is a beautiful construction, I wonder if this really is necessary. Is […]

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?

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

Could you please explain me the reason why they are isomorphic? Thanks, bye!

The last accidental spin groups

For dimensions $n\le 6$ there are accidental isomorphisms of spin groups with other Lie groups: $\DeclareMathOperator{Spin}{\mathrm{Spin}}$ $$\begin{array}{|l|l|} \hline \Spin(1) & \mathrm{O}(1) \\ \hline \Spin(2) & \mathrm{SO}(2) \\ \hline \Spin(3) & \mathrm{Sp}(1) \\ \hline \Spin(4) & \mathrm{Sp}(1)\times\mathrm{Sp}(1) \\ \hline \Spin(5) & \mathrm{Sp}(2) \\ \hline \Spin(6) & \mathrm{SU}(4) \\ \hline \end{array} $$ The definition of $\Spin(n)$ is […]

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

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