Group Isomorphisms Review

# Group Isomorphisms Review

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

- On the
**Group Isomorphisms**page 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:

\begin{align} \quad f(x *_1 y) = f(x) *_2 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 $(G_1, *_1)$ 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 $G_1 \not \cong G_2$. |

- 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 $(G_1, *_1)$ 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 $G_3$. |

- 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 $(G_1, *_1)$ and $(G_2, *_2)$ are groups with identities $e_1$ and $e_2$ respectively, and $G_1 \cong G_2$ with isomorphism $f : G_1 \to G_2$ then:

\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 $f(x^{-1}) \in G_2$.

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

Theorem |
---|

(a) If $(G_1, *_1)$ 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.

- 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