Group Isomorphisms Review

We will now review some of the recent material regarding group isomorphisms.

• On the Group Isomorphismspage we said that two groups $(G, \cdot)$ and $(G, *)$ are Isomorphic denoted $G \cong H$ if there exists a function $f : G \to H$ called an Isomorphism between these groups with the following properties:
  • (1) $f$ is bijective
  • (2) For all $x, y \in G$ we have that:

(1)

\begin{align} \quad f(x \cdot y) = f(x) * f(y) \end{align}

• On the Necessary Conditions for Two Groups to Be Isomorphic page we looked at some important results regarding two groups being isomorphic. These results are summarized below:

Theorem
(a) If $(G, \cdot)$ and $(H, *)$ are finite groups and $G \cong H$ then $|G| = |H|$.
(b) If $(G, \cdot)$ is a finite group and $(H, *)$ is an infinite group then $G \not \cong H$.
(c) If $(G, \cdot)$ is an abelian group and $(H, *)$ is a non-abelian group then $G \not \cong H$.

• We looked at some more results regarding group isomorphisms on the Basic Theorems Regarding Group Isomorphisms page. These results are summarized below.

Theorem
(a) If $(G, \cdot)$ and $(H, *)$ are groups and $f : G \to H$ is an isomorphism from $G$ to $H$ then $f^{-1} : H \to G$ is an isomorphism from $H$ to $G$.
(b) If $(G, \cdot)$, $(H, *)$, $(G_3, *_3)$ are groups such that $G \cong H$, $H \cong G_3$, and $f : G \to H$ and $g : H \to G_3$ are isomorphisms, then $G \cong G_3$ and $g \circ f : G \to G_3$ is a corresponding isomorphism from $G$ to $G_3$.

• On the Preservation of the Identities and Inverses under Group Isomorphisms we saw that group isomorphisms preserve a lot of the structure between groups. We saw that if $(G, \cdot)$ and $(H, *)$ are groups with identities $e_1$ and $e_2$ respectively, and $G \cong H$ with isomorphism $f : G \to H$ then:

(2)

\begin{align} \quad f(e_1) = e_2 \end{align}

• In other words, isomorphisms map the identity element in the domain group to the identity element in the codomain group.

• Furthermore, we saw that if $x \in G$ has inverse element $x^{-1} \in G$ then $f(x) \in H$ has inverse element $f(x^{-1}) \in H$.

• On the Preservation of Special Properties under Group Isomorphisms page we resummarized some special properties that are preserved between isomorphic groups:

Theorem
(a) If $(G, \cdot)$ and $(H, *)$ are isomorphic and $(G, \cdot)$ is an abelian group then $(H, *)$ is an abelian group.
(b) If $(G, \cdot)$ and $(H, *)$ are isomorphic and $(G, \cdot)$ is a cyclic group then $(H, *)$ is a cyclic group.
(c) If $(G, \cdot)$ and $(H, *)$ are isomorphic and $|G| = n$ then $|H| = n$.

• On the Cyclic Groups and their Isomorphisms page we proved a very important result. We proved that every infinite cyclic group is isomorphic to $\mathbb{Z}$, and that every finite cyclic group of order $n$ is isomorphic to $\mathbb{Z}_n$.

• We also proved that if $(G, *)$ is a group with $|G| = p$ where $p$ is prime, then $G$ is isomorphic to $\mathbb{Z}_p$. This is because if $|G| = p$ then $G$ must be cyclic.

• On the Cayley's Group Isomorphism Theorem page we proved Cayley's Group Isomorphism theorem which states that every group is isomorphic to a group of permutations