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Section Daily Prep 7
You will explore some consequences of Lagrange’s Theorem, learn the definition of the external direct product of groups, and begin to practice doing computations in an external direct product.
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 the following mathematical results: Order of an Element Divides Order of the Group, Groups of Prime Order are Cyclic
State and instantiate the definition of: external direct product
Compute products and orders of elements in an external direct product of groups
Subsection Resources for Learning
Use these resources to prepare for class and answer the questions below.
Gallian, Chapter 8, pp. 156-157
Review Ch. 7 and the class notes as needed.
Figure 71. Reference Video for Corollaries of Lagrange’s Theorem
Figure 72. Reference Video for Direct Products
Subsection Important Terms
Corollary 74 . Order of an Element Divides Order of the Group.
Let
\(G\) be a finite group,
\(g \in G\text{,}\) and
\(H\leq G\text{.}\) Then the order of
\(g\) and the order of
\(H\) divide the order of
\(G\text{.}\)
Corollary 75 . Groups of Prime Order are Cyclic.
Let
\(G\) be a group of prime order
\(p\text{.}\) Then
\(G\) is cyclic, generated by any non-identity element.
Definition 76 . External Direct Product of Groups.
Let \(G\) and \(H\) be groups. The external direct product of \(G\) and \(H\text{,}\) denoted \(G \times H\text{,}\) is the set of all ordered pairs \((g, h)\) with \(g \in G\) and \(h \in H\text{,}\) equipped with the operation defined component-wise by
\begin{equation*}
(g_1, h_1)(g_2, h_2) = (g_1 g_2, h_1 h_2)\text{.}
\end{equation*}
The external product of \(n\) groups \(G_1, G_2, \ldots, G_n\) is defined similarly.
