Tour:Introduction four (beginners)
This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)
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General instructions for the tour | Pedagogical notes for the tour
In part one of the tour, we focused on basic definitions: group, subgroup, trivial group, and Abelian group. In part two, we tried to understand how to use the associativity, identity element and inverses to perform elementary manipulations in groups. In part three, we began understanding the subgroup structure of groups, the rules about intersections, unions and subgroups generated, and the notions of left and right coset.
This part of the tour is a preliminary look at some important classes of examples of groups, specifically, cyclic groups. We also pack here some general tools and approaches that will be useful later on.
- Multiplication table of a finite group: A tabular representation of the multiplication rule of a finite group.
- Isomorphism of groups: A straightforward definition of what an equality of two groups would mean.
- Isomorphic groups: A straightforward definition of what it means for two groups to be equal.
- Group of integers
- Group of integers modulo n
- Order of an element
- Cyclic group: A cursory definition of cyclic group.
- Equivalence of definitions of cyclic group
- Every nontrivial subgroup of the group of integers is cyclic on its smallest element
- Subgroup containment relation equals divisibility relation on generators
- No proper nontrivial subgroup implies cyclic of prime order
- Exploration of cyclic groups: A survey article looking at cyclic groups, how they are constructed, and a number of interesting facts about them.
- Cyclicity is subgroup-closed
- Multiplicative group modulo n
- Elements of multiplicative group equal generators of additive group
- Multiplicative group modulo a prime is cyclic
Prerequisites for this part: Content covered in parts one, two, and three (or equivalent content). In particular, the definitions of group, subgroup, trivial group, Abelian group, identity element, inverses, intersection of subgroups, join of subgroups, generating set of a group, left coset of a subgroup. Also, the major facts proved about these.
Goal of this part: The goal here is a preliminary study an important class of groups: the cyclic groups. We study these from the viewpoint of how they occur naturally, and from the viewpoint of the generic tools we've developed for handling groups and subgroups.
This page is part of the Groupprops Guided tour for beginners (Jump to beginning of tour). If you found anything difficult or unclear, make a note of it; it is likely to be resolved by the end of the tour.
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