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

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?

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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