The lecture notes I'm working through pose an exercise to find groups that are isomorphic to each other or to subgroups of other groups. The groups listed so far are:
- The trivial group $\{e\}$
- Numerical systems under addition: $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$ with their usual addition.
- Any group with two elements $\{e,x\}$ (with the only possible multiplication table: $ex = xe = x$, $ee = e$, and $xx = e$.
- $\mathbb{Z}/n = \{0,1, \ldots, n-1\}$ with addition mod $n$
- Non-zero elements $\mathbb{Q}^{*}$, $\mathbb{R}^{*}$, $\mathbb{C}^{*}$ with multiplication
- $S^1$, the complex unit circle in $\mathbb{C}$
- Permutations of any set $A$, denoted $\mathrm{Perm}(A)$
- $S_n$, the symmetric group on $n$ letters (the above with $A = \{1, \ldots, n\}$
- The general linear group and special linear group of matrices
- Products, finite and infinite, of groups
- Direct sums of groups
- $\mathbb{R}/\mathbb{Z} = [0,1)$ with addition mod $1$
My goal is to find as many isomorphisms as I can between these groups, including subgroups thereof. That is, if $H \lhd G$, then $H$ is isomorphic to a subgroup of $G$. There is only one group of order $1$ and one group of order $2$, up to isomorphism, so I don't think I'm missing any with those. As $\mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}$, I can say that
- $(\mathbb{Z}, +) \lhd (\mathbb{Q}, +) \lhd (\mathbb{R}, +) \lhd (\mathbb{C}, +)$
and, since the integers do not form a multiplicative group:
- $(\mathbb{Q}^{*}, \times) \lhd (\mathbb{R}^{*}, \times) \lhd (\mathbb{C}^{*}, \times)$
Taking $A$ to be any $n$-element set, we can say that:
- $\mathrm{Perm}(A) \cong S_n$
As $\mathbb{R} \subset \mathbb{C}$, we can say that:
- $\mathrm{GL}_n (\mathbb{R}) \lhd \mathrm{GL}_n (\mathbb{C})$
As $S^1 : = \{z \in \mathbb{C} \mid |z| = 1\} \subset \mathbb{C}^{*}$, we have
- $(S^1, \times) \subset (\mathbb{C}^{*}, \times)$
By definition, a direct sum is an "infinite tuple" $(a_1, a_2, \ldots, )$ where $a_i \in G_i$ and all but finitely many terms are the identity. A direct product lifts the latter restriction, so any direct sum is a subgroup of the corresponding direct product:
- $\bigoplus G_i \lhd \prod G_i$
For $1 \leq i \leq n$, if $H_i \lhd G_i$, then every tuple $(h_1, \ldots, h_n)$ of elements of $H_i$ is an element of $\prod\limits_{i=1}^n G_i$, so we have:
- $\forall 1 \leq i \leq n, \; H_i \lhd G_i \implies \prod\limits_{i=1}^n H_i \lhd \prod\limits_{i=1}^n G_i$.
If $F \cong F'$ as fields, then I should be able to write down an isomorphism between $\mathrm{GL}_n (F)$ and $\mathrm{GL}_n (F')$, where the map sends an element $A \in \mathrm{GL}_n (F)$ to its image, entry-by-entry, under the isomorphism between $F$ and $F'$.
- $\mathrm{GL}_n (F) \cong \mathrm{GL}_n (F')$
The $n$th roots of unit, $\{z \in \mathbb{C} \mid |z|^n = 1\}$ form a cyclic subgroup of $\mathbb{C}^{\star}$, and any cyclic subgroup of size $n$ is isomorphic to $\mathbb{Z}/n$:
- $\{z \in \mathbb{C} \mid |z|^n = 1\} \cong \mathbb{Z}/n$.
I'm not able to think of any others, though there's a hint in the notes to consider $\mathbb{Z}/n$, $\mathbb{C}^{*}$, and $\mathrm{GL}_2 (\mathbb{R})$. I'd appreciate if someone could help with this, either in providing feedback on what I wrote above or in pointing out some isomorphisms involving these above three groups or isomorphisms I missed.