• Home
  • Textbooks
  • A Book of Abstract Algebra
  • ISOMORPHISM

A Book of Abstract Algebra

Charles C. Pinter

Chapter 9

ISOMORPHISM - all with Video Answers

Educators

Section 1

A

04:16

Problem 1

The following three facts about isomorphism are true for all groups:
1. Every group is isomorphic to itself.
2. If $G_{1} \cong G_{2}$, then $G_{2} \cong G_{1}$.
3. If $G_{1} \cong G_{2}$ and $G_{2} \cong G_{3}$, then $G_{1} \cong G_{3}$.
Fact 1 asserts that for any group $G$, there exists an isomorphism from $G$ to $G$.
Fact 2 asserts that, if there is an isomorphism $f$ from $G_{1}$ to $G_{2}$, there must be some isomorphism from $G_{2}$ to $G_{1}$. Well, the inverse of $f$ is such an isomorphism.
Fact 3 asserts that, if there are isomorphisms $f: G_{1} \rightarrow G_{2}$ and $g: G_{2} \rightarrow G_{3}$, there must be an isomorphism from $G_{1}$ to $G_{3}$. One can easily guess that $g \circ f$ is such an isomorphism. The details of facts 1,2 , and 3 are left as exercises.
Let $G$ be any group. If $\varepsilon: G \rightarrow G$ is the identity function, $\varepsilon(x)=x$, show that $\varepsilon$ is an isomorphism.

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
01:02

Problem 2

The following three facts about isomorphism are true for all groups:
1. Every group is isomorphic to itself.
2. If $G_{1} \cong G_{2}$, then $G_{2} \cong G_{1}$.
3. If $G_{1} \cong G_{2}$ and $G_{2} \cong G_{3}$, then $G_{1} \cong G_{3}$.
Fact 1 asserts that for any group $G$, there exists an isomorphism from $G$ to $G$.
Fact 2 asserts that, if there is an isomorphism $f$ from $G_{1}$ to $G_{2}$, there must be some isomorphism from $G_{2}$ to $G_{1}$. Well, the inverse of $f$ is such an isomorphism.
Fact 3 asserts that, if there are isomorphisms $f: G_{1} \rightarrow G_{2}$ and $g: G_{2} \rightarrow G_{3}$, there must be an isomorphism from $G_{1}$ to $G_{3}$. One can easily guess that $g \circ f$ is such an isomorphism. The details of facts 1,2 , and 3 are left as exercises.
Let $G_{1}$ and $G_{2}$ be groups, and $f: G_{1} \rightarrow G_{2}$ an isomorphism. Show that $f^{-1}$ : $G_{2} \rightarrow G_{1}$ is an isomorphism. [HINT: Review the discussion of inverse functions at the end of Chapter 6. Then, for arbitrary elements $a, b \in G_{1}$, let $c=f(a)$ and $d=f(b)$. Note that $a=f^{-1}(c)$ and $b=f^{-1}(d)$. Show that $\left.f^{-1}(c d)=f^{-1}(c) f^{-1}(d) .\right]$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:02

Problem 3

The following three facts about isomorphism are true for all groups:
1. Every group is isomorphic to itself.
2. If $G_{1} \cong G_{2}$, then $G_{2} \cong G_{1}$.
3. If $G_{1} \cong G_{2}$ and $G_{2} \cong G_{3}$, then $G_{1} \cong G_{3}$.
Fact 1 asserts that for any group $G$, there exists an isomorphism from $G$ to $G$.
Fact 2 asserts that, if there is an isomorphism $f$ from $G_{1}$ to $G_{2}$, there must be some isomorphism from $G_{2}$ to $G_{1}$. Well, the inverse of $f$ is such an isomorphism.
Fact 3 asserts that, if there are isomorphisms $f: G_{1} \rightarrow G_{2}$ and $g: G_{2} \rightarrow G_{3}$, there must be an isomorphism from $G_{1}$ to $G_{3}$. One can easily guess that $g \circ f$ is such an isomorphism. The details of facts 1,2 , and 3 are left as exercises.
Let $G_{1}, G_{2}$, and $G_{3}$ be groups, and let $f: G_{1} \rightarrow G_{2}$ and $g: G_{2} \rightarrow G_{3}$ be isomorphisms. Prove that $g \circ f: G_{1} \rightarrow G_{3}$ is an isomorphism.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator