# Tour:Introduction five (beginners)

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This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)PREVIOUS: Inquiry problems four (beginners)|NEXT: Symmetric groupPREVIOUS SECTION: Introduction four|NEXT SECTION: Introduction six

General instructions for the tour | Pedagogical notes for the tour

Hopefully, by the time you've reached this part of the tour, you have under your belt the basic definitions of group, subgroup, trivial group, and abelian group, and you are familiar with notions such as joins, intersections, and coset spaces.

In part four, you saw some of the easiest examples of groups: the cyclic groups. Cyclic groups are all abelian, and the structure and behavior of cyclic groups is very different from that of most other groups. In this part, we look at another very important class of groups: the symmetric groups. We will also encounter many general constructions involving groups.

The following pages comprise part five:

- symmetric group: A quick definition of the symmetric group on a set, along with an explanation of the two-line notation and the one-line notation.
- Understanding the cycle decomposition: A survey article with a gentle introduction to cycle decompositions, along with many examples.
- Cycle decomposition for permutations: A precise definition of the cycle decomposition of a permutation.
- Subset containment gives inclusion of symmetric groups: A quick definition-cum-observation.
- Group action: A quick definition.
- Homomorphism of groups: A quick definition.
- Equivalence of definitions of group homomorphism
- Understanding the definition of a homomorphism: A survey article exploring the concept of a homomorphism.
- Equivalence of definitions of group action
- Cayley's theorem
- External direct product
- Product of subgroups
- Permuting subgroups
- Equivalence of definitions of permuting subgroups
- Permutable complements
- Every group of given order is a permutable complement for symmetric groups
- Conjugation as a relabeling of the underlying set: A survey article that introduces the notion of conjugation along with its natural importance for group actions and the symmetric group.