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Section Daily Prep 3
You will review some of the properties of cyclic subgroups generated by an element of a group and the center of a group from class on Thursday in preparation for our deeper discussion of cyclic 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.
Determine the cyclic subgroup generated by an element of a group.
State and instantiate the definitions of: order of a group, order of an element, subgroup, cyclic subgroup of a group generated by an element, center of a group
State and instantiate the definition of a cyclic group and a generator of a group.
Subsection Resources for Learning
Use these resources to prepare for class and answer the questions below.
Gallian, Chapter 4, pg.77 and the top of pg. 78
Review Ch. 3 and the class notes as needed.
Figure 24. Reference Video for Cyclic Groups
Subsection Important Terms
Definition 25 . Cyclic Group.
A group
\(G\) is called
cyclic if there is an element
\(a\in G\) such that
\(G=\langle a \rangle=\{a^n\mid n\in\Z\}\text{.}\) The element
\(a\) is called a
generator of the group.
