Examples of non-obvious isomorphisms following from the first isomorphism theorem

Question

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 $$x+\mathbb{Z}\mapsto e^{2\pi x i}$$

• $\dfrac{\mathbb Z \times \mathbb Z}{\langle (m,n)\rangle}\cong\mathbb Z$, where $m,n$ are integers.

• $\dfrac{\mathbb R^\star}{\{1, -1\}} \cong \mathbb R^+$.

I would like to see more examples of such isomorphisms, intended both as a reference and to help me study for the test! Thank you.

Answer 1

Sometimes, instead of using the first isomorphism theorem as a tool to construct isomorphisms, it can be used as a tool to construct subgroups with certain properties. For example, consider the problem:

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Let $G$ be a finite group with subgroup $H$, $[G:H] = n$, then $H$ contains a normal subgroup of index $\leq n!$

Solution: $G$ acts on $G/H$ by $x(gH) = (xg)H$. This induces a homomorphism $\phi$ from $G$ to $\Sigma_{G/H}$. Suppose that $x$ is in the kernel of $\phi$. Then $xH = H$, so that $x \in H$. Thus the kernel $K$ of $\phi$ is contained inside $H$. By the first isomorphism theorem, $G/K$ is isomorphic to a subgroup of $\Sigma_{G/H}$, which necessarily has order less than $|\Sigma_{G/H}| = n!$.

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The purpose of this example is to demonstrate that you can produce a non-obvious subgroup by picking an appropriate homomorphism, looking at its kernel, and applying the first isomorphism theorem.

Answer 2

I really like isomorphisms involving matrix groups. For example, $SU(2)/\{\pm I\}\simeq SO(3)$ or $SL_2(\mathbb{F}_5)/\{\pm I\}\simeq A_5$. Here $SU(2)$ is the group of unitary $2\times 2$ matrices, $SO(3)$ is the group of real orthogonal $3\times 3$ matrices, $SL_2(\mathbb{F}_5)$ are matrices with coefficiens in the field with $5$ elements with $\det=1$, and $A_5$ is a group of even permutations of $5$ elements.

I am not sure if this is exactly what you are asking for, but I hope it helps a bit.