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Section Daily Prep 6
You will begin to explore the idea of the cosets of a subgroup in a group. We will see that these partition a group and give us more understanding about the structure of that group.
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: left coset of
\(H\) in
\(G\text{,}\) right coset of
\(H\) in
\(G\)
Given a subgroup
\(H\) of a group
\(G\text{,}\) compute the left (or right) cosets of
\(H\) in
\(G\text{.}\)
Subsection Resources for Learning
Use these resources to prepare for class and answer the questions below.
Gallian, Chapter 7, pp. 144-145
Review Ch. 6 and the class notes as needed.
Figure 58. Reference Video for Cosets & Cayley Diagrams
Subsection Important Terms
Definition 59 . Cosets of a Subgroup.
Let \(G\) be a group and \(H\leq G\text{.}\) For any \(a \in G\text{,}\) the left coset of \(H\) in \(G\) containing \(a\) is the set
\begin{equation*}
aH = \{ah \mid h \in H\}\text{.}
\end{equation*}
The right coset of \(H\) in \(G\) containing \(a\) is the set
\begin{equation*}
Ha = \{ha \mid h \in H\}\text{.}
\end{equation*}
The element \(a\) is called the coset representative of \(aH\) (or \(Ha\) ) .
