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 they are isomorphic or not?

I don't know if this is really amazing, but I was quite surprised when I did discover these isomorphisms:

• $\mathbb R^n$ and $\mathbb R^m$ are isomorphic as abelian groups (where $n,m \geq 1$). In fact, pick any Hamel basis of $\mathbb R^n$ and $\mathbb R^n$ as $\mathbb Q$ vector space. Both basis are in bijection with $\mathbb R$, so any bijection between them give an isomorphism of $\mathbb Q$ vector spaces, in particular of abelian groups.
• A really surprising example (for me at least): the free group with countably many generator is isomorphic to a subgroup of $F_2$, the free group with two generators! This fact comes from the covering $\sin : X \to X'$ where $X = \mathbb C \setminus \{\frac{\pi}{2} + k\pi, k \in \mathbb Z\}, X' = \mathbb C \setminus \{-1,1\}$, and the more general fact that for any covering between nice space the induced maps on fundamental groups is injective.
• $L^2(\mathbb S^1) \cong \ell^2(\mathbb Z)$ via Parseval equality.
• The space $\mathcal M_k$ of modular forms of weight $2k$ is finite dimensional (which is already non-trivial) and moreover, if $\mathcal M = \bigoplus_{k \in \mathbb N} \mathcal M_k$, we have the a graded algebra isomorphism $\mathcal M \cong \mathbb C[x,y] $ where $x$ has degree $2$ and $y$ has degree $3$ (they represents Eisenstein series $G_4(z)$ and $G_6(z)$ respectively). This is quite surprising that the algebra of all modular forms is simply isomorphic to an algebra of polynomials !
• Finally, a cute example: the group of direct isometries which preserve a cube are isomorphic to the symmetric group $\mathfrak S_4$. In fact, this is exactly the permutation group of the big diagonals $d_1,d_2,d_3,d_4$ of the cube.

My favorite, and a classical example, is the Outer automorphism of $S_6$, which loosely arises from the 'coincidence' that ${6\choose 2}=15=\frac{1}{2^3}\frac{6!}{3!}$ — that is, the number of (unordered) pairs of elements of $\{1,2,3,4,5,6\}$ is exactly the same as the number of partitions into three pairs. John Baez has a nice essay that offers more details on exactly how the automorphism can be defined.

One of my favourites (and certainly one of the things that introduced me to the concept of isomorphism, and perhaps an obvious one) is the isomorphism of $\Bbb C$ and $(M, \times)$, where $$M = \left\lbrace \begin{pmatrix}a& -b \\ b &a \end{pmatrix} : a, b \in \Bbb R, a^2 + b^2 \neq 0\right\rbrace$$

given by $\varphi:\Bbb C\setminus\{ 0 \} \to (M, \times)$, with $(a + bi) \mapsto \begin{pmatrix}a &-b\\ b & a \end{pmatrix}.$

You learn early on that the complex numbers are geometric in nature, and you also learn that matrices encode geometric features. This was a nice connection for me and was in fact one of the things that got me interested in mathematics in the first place!

Consider groups $\mathrm{PSL}_2(\mathbb{F}_p)=\mathrm{SL}_2(\mathbb{F}_p)/Z$, $Z$ being the center of the group. Then $$\mathrm{PSL}_2(\mathbb{F}_4)\cong \mathrm{PSL}_2(\mathbb{F}_5).$$ Another example, consider commutator subgroup of $\mathrm{SL}_2(\mathbb{Z})$, it is isomorphic to free group on $2$ letters.