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Section Daily Prep 9
You will practice more with quotient groups and learn the definition of a homomorphism of groups.
Objectives: Basic Learning Objectives
Before our class meeting, you should use the resources below to be able to learn the following. You should be reasonably fluent with these; we’ll answer some questions on them in class but not reteach them in detail.
Perform computations in the quotient group
\(G/H\) for a normal subgroup
\(H\) in
\(G\)
State and instantiate the definition of: homomorphism, kernel
Given a map between groups, determine if it is a homomorphism and if so compute its kernel
Subsection Resources for Learning
Use these resources to prepare for class and answer the questions below.
Gallian, Chapter 10, pp. 194-195
Review Ch. 9 and the class notes as needed
Figure 96. Reference Video for Homomorphisms
Subsection Important Terms
Definition 97 . Group Homomorphism.
A homomorphism from a group \(G\) to a group \(\overline{G}\) is a map \(\phi: G \to \overline{G}\) that preserves the group operation: for all \(a,b\in G\) we have
\begin{equation*}
\phi(ab)=\phi(a)\phi(b)\text{.}
\end{equation*}
Definition 98 . Kernel of a Homomorphism.
The kernel of a homomorphism \(\phi: G \to \overline{G}\) is the set of elements of \(G\) that map to the identity element of \(\overline{G}\text{:}\)
\begin{equation*}
\ker(\phi) = \{x\in G : \phi(x) = e_{\overline{G}}\}\text{.}
\end{equation*}
