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Section Daily Prep 5
You will begin to explore the idea of an
isomorphism of groups - the concept of “sameness” for 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.
State and instantiate the definitions of: isomorphism, isomorphic groups
Given a map between groups, determine if it is an isomorphism
Subsection Resources for Learning
Use these resources to prepare for class and answer the questions below.
Gallian, Chapter 6, pp. 127-129
Review Ch. 5 and the class notes as needed.
Figure 46. Reference Video for Isomorphisms
Subsection Important Terms
Definition 47 . Group Isomorphism.
Let \(G, G'\) be groups. A map \(\varphi:G\to G'\) is an isomorphism of groups if it is
operation preserving: for all \(a,b\in G\) we have
\begin{equation*}
\varphi(a \circ_G b)=\varphi(a)\circ_{G'}\varphi(b)\text{.}
\end{equation*}
If such a map exists then we write \(G\isom G'\) and say that \(G\) and \(G'\) are isomorphic .
