Group Isomorphisms Review

Group Isomorphisms Review

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

  • On the Group Isomorphismspage we said that two groups $(G, *_1)$ and $(G, *_2)$ are Isomorphic denoted $G_1 \cong G_2$ if there exists a function $f : G_1 \to G_2$ called an Isomorphism between these groups with the following properties:
    • (1) $f$ is bijective
    • (2) For all $x, y \in G_1$ we have that:
(1)
\begin{align} \quad f(x *_1 y) = f(x) *_2 f(y) \end{align}
Theorem
(a) If $(G_1, *_1)$ and $(G_2, *_2)$ are finite groups and $G_1 \cong G_2$ then $|G_1| = |G_2|$.
(b) If $(G_1, *_1)$ is a finite group and $(G_2, *_2)$ is an infinite group then $G_1 \not \cong G_2$.
(c) If $(G_1, *_1)$ is an abelian group and $(G_2, *_2)$ is a non-abelian group then $G_1 \not \cong G_2$.
Theorem
(a) If $(G_1, *_1)$ and $(G_2, *_2)$ are groups and $f : G_1 \to G_2$ is an isomorphism from $G_1$ to $G_2$ then $f^{-1} : G_2 \to G_1$ is an isomorphism from $G_2$ to $G_1$.
(b) If $(G_1, *_1)$, $(G_2, *_2)$, $(G_3, *_3)$ are groups such that $G_1 \cong G_2$, $G_2 \cong G_3$, and $f : G_1 \to G_2$ and $g : G_2 \to G_3$ are isomorphisms, then $G_1 \cong G_3$ and $g \circ f : G_1 \to G_3$ is a corresponding isomorphism from $G_1$ to $G_3$.
(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 $x \in G_1$ has inverse element $x^{-1} \in G_1$ then $f(x) \in G_2$ has inverse element $f(x^{-1}) \in G_2$.
Theorem
(a) If $(G_1, *_1)$ and $(G_2, *_2)$ are isomorphic and $(G_1, *_1)$ is an abelian group then $(G_2, *_2)$ is an abelian group.
(b) If $(G_1, *_1)$ and $(G_2, *_2)$ are isomorphic and $(G_1, *_1)$ is a cyclic group then $(G_2, *_2)$ is a cyclic group.
(c) If $(G_1, *_1)$ and $(G_2, *_2)$ are isomorphic and $|G_1| = n$ then $|G_2| = n$.
  • 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.
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