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: States that an arbitrary intersection of subgroups is a subgroup
• Union of two subgroups is not a subgroup: States that a union of two subgroups isn't a subgroup unless one of them is inside the other.
• Left coset of a subgroup: Defines the notion of left coset of a subgroup.
• Left cosets partition a group: Explains how the left cosets of a subgroup partition the group.
• Left cosets are in bijection via left multiplication: Explains that the left cosets of a subgroup are in bijection.
• Right coset of a subgroup: Defines the notion of right coset of a subgroup.
• Left and right coset spaces are naturally isomorphic: Explains how the collection of left cosets and the collection of right cosets can be identified with each other.
• Index of a subgroup: Defines the notion of index of a subgroup in a group.
• Lagrange's theorem: A theorem stating that for a finite group, the order of a subgroup must divide the order of the group.
• Generating set of a group: Defines generating set.
• Subgroup generated by a subset: Defines the notion of subgroup generated by a subset.
• Join of subgroups: Defines the notion of subgroup generated or join.

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

Goal of this part:

• 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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