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 […]

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

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 $PGL_2(5)\cong S_5$? And is there a set of 5 elements on which $PGL_2(5)$ acts?

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 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?

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

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 […]

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

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