Tour:Introduction three (beginners): Difference between revisions

Revision as of 19:31, 29 June 2008

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:

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.
PREVIOUS: Mind's eye test two (beginners) | UP: Getting started (beginners) | NEXT: Intersection of subgroups is subgroup

@@ Line 6: / Line 6: @@
 In this part, we'll see:
-* [[Tour:Intersection of subgroups is subgroup|Intersection of subgroups is subgroup]]: States that an arbitrary intersection of subgroups is a subgroup
+* [[Tour:Intersection of subgroups is subgroup|Intersection of subgroups is subgroup]]: A short, ''mind's eye'' proof
-* [[Tour:Union of two subgroups is not 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.
+* [[Tour:Union of two subgroups is not a subgroup|Union of two subgroups is not a subgroup]]: A somewhat long, but essentially ''mind's eye'' proof
-* [[Tour:Left coset of a subgroup|Left coset of a subgroup]]: Defines the notion of left coset of a subgroup.
+* [[Tour:Left coset of a subgroup|Left coset of a subgroup]]: A simple definition.
-* [[Tour:Left cosets partition a group|Left cosets partition a group]]: Explains how the left cosets of a subgroup partition the group.
+* [[Tour:Left cosets partition a group|Left cosets partition a group]]: A short, ''mind's eye'' proof.
-* [[Tour:Left cosets are in bijection via left multiplication|Left cosets are in bijection via left multiplication]]: Explains that the left cosets of a subgroup are in bijection.
+* [[Tour:Left cosets are in bijection via left multiplication|Left cosets are in bijection via left multiplication]]: A short, ''mind's eye'' proof.
-* [[Tour:Right coset of a subgroup|Right coset of a subgroup]]: Defines the notion of right coset of a subgroup.
+* [[Tour:Right coset of a subgroup|Right coset of a subgroup]]: A simple definition.
-* [[Tour:Left and right coset spaces are naturally isomorphic|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.
+* [[Tour:Left and right coset spaces are naturally isomorphic|Left and right coset spaces are naturally isomorphic]]: A short, ''mind's eye'' proof.
-* [[Tour:Index of a subgroup|Index of a subgroup]]: Defines the notion of index of a subgroup in a group.
+* [[Tour:Index of a subgroup|Index of a subgroup]]: A simple definition.
-* [[Tour:Lagrange's theorem|Lagrange's theorem]]: A theorem stating that for a finite group, the order of a subgroup must divide the order of the group.
+* [[Tour:Lagrange's theorem|Lagrange's theorem]]: An important result for finite groups, with a short, ''mind's eye'' proof.
-* [[Tour:Generating set of a group|Generating set of a group]]: Defines generating set.
+* [[Tour:Generating set of a group|Generating set of a group]]: A simple definition.
-* [[Tour:Subgroup generated by a subset|Subgroup generated by a subset]]: Defines the notion of subgroup generated by a subset.
+* [[Tour:Subgroup generated by a subset|Subgroup generated by a subset]]: A simple definition.
-* [[Tour:Join of subgroups|Join of subgroups]]: Defines the notion of ''subgroup generated'' or ''join''.
+* [[Tour:Join of subgroups|Join of subgroups]]: A simple definition of ''subgroup generated'' or ''join''.
+We'll also see some consolidation pages:
+* [[Tour:Factsheet three (beginners)|Factsheet three]]
+* [[Tour:Mind's eye test three (beginners)|Mind's eye test three]]
+* [[Tour:Confidence aggregator three (beginners)|Confidence aggregator three]]
 <section end="pagelist"/>
@@ Line 23: / Line 29: @@
 <section end="prerequisite"/>
-<section begin="goal"/>'''Goal of this part''':
+<section begin="goal"/>'''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?