Tour:Introduction three (beginners)

This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)
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In part one of the guided tour, we focused on some basic definitions: group, subgroup, trivial group, and Abelian group. In part two, we focused on some basic manipulations involving groups (not necessarily Abelian). Here, in part three, we work to understand subgroup structure somewhat better.

In this part, we'll see:

• Intersection of subgroups is subgroup: A short, mind's eye proof
• Union of two subgroups is not a subgroup: A somewhat long, but essentially mind's eye proof
• Left coset of a subgroup: A simple definition.
• Left cosets partition a group: A short, mind's eye proof.
• Left cosets are in bijection via left multiplication: A short, mind's eye proof.
• Right coset of a subgroup: A simple definition.
• Left and right coset spaces are naturally isomorphic: A short, mind's eye proof.
• Index of a subgroup: A simple definition.
• Lagrange's theorem: An important result for finite groups, with a short, mind's eye proof.
• Generating set of a group: A simple definition.
• Subgroup generated by a subset: A simple definition.
• Join of subgroups: A simple definition of subgroup generated or join.

We'll also see some consolidation pages:

• Factsheet three
• Mind's eye test three
• Confidence aggregator three
• Interdisciplinary problems three
• Examples peek three

Prerequisites for this part: Parts one and two (or equivalent content)

Goal of this part: We'll seek answers to the questions:

• What can we say about set-theoretic operations done on subgroup (like unions and intersections)?
• How does the nature of a group control the nature of possible subgroups?
• What is special about finite groups and subgroups thereof?

Some general suggestions/motivation for this part: If you've gone through parts one and two of the tour, you have seen some important basic definitions (group, subgroup, trivial group and Abelian group) and have also dabbled with simple manipulations involving groups. In this part of the tour, we focus on using these manipulation techniques to understand a few things about the structure of subgroups of a group.

At this stage, you will be asked to take a more proactive role in the tour. Often, multiple definitions of the same concept will be presented, and you will need to work out why these definitions are equivalent. The definitions may be only slightly different in wording, and their equivalence may stem from some of the simple manipulations seen in part two. The WHAT YOU NEED TO DO on the relevant pages will strongly encourage you to attempt to prove the equivalence of definitions.

Similarly, the proofs of some facts presented to you may implicitly use ideas involving manipulations in groups, and you are expected to convince yourself of the reasoning for individual steps.

If you find yourself uncomfortable with either the statements or the proofs of these facts, you are encouraged to return to part two, review the important facts, and review the Mind's Eye Test.

In addition to understanding the statements and their proofs, you should also seek to understand the overall direction of the results. For instance, on seeing a new result about intersections of subgroups, you should pause and ask yourself: what was already known, and how does this new result fit in? The design of the tour is such that it will prompt you to think along these lines, but the more proactive you are in this regard, the faster your learning.

Review the three questions stated in the goal above, and keep your eyes open for how the results we obtain help in answering these questions.

This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)
PREVIOUS: Objective evaluator two (beginners)| NEXT: Intersection of subgroups is subgroup
PREVIOUS SECTION: Introduction two|NEXT SECTION: Introduction four
General instructions for the tour | Pedagogical notes for the tour