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Section Daily Prep 8
You will learn the definition of a normal subgroup and the Normal Subgroup Test, and begin practicing determining whether a given subgroup of a group is normal.
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 definition of: normal subgroup
State the following mathematical result: Normal Subgroup Test
Subsection Resources for Learning
Use these resources to prepare for class and answer the questions below.
Gallian, Chapter 9, pp. 185-186
Review Ch. 8 and the class notes as needed.
Figure 86. Reference Video for Normal Subgroups
Subsection Important Terms
Definition 87 . Normal Subgroup.
A subgroup
\(N\) of of a group
\(G\) is
normal in
\(G\) if
\(gN=Ng\) for all
\(g\in G\text{.}\) We denote this as
\(N \normaleq G\text{.}\)
Theorem 88 . Normal Subgroup Test.
Let
\(N\) be a subgroup of a group
\(G\text{.}\) Then
\(N\) is normal in
\(G\) if and only if
\(xNx^{-1} \subseteq N\) for all
\(x\in G\text{.}\)
