Difference between revisions of "Tour:Index of a subgroup"

From Groupprops
Jump to: navigation, search
Line 3: Line 3:
 
{{quotation|'''WHAT YOU NEED TO DO''':  
 
{{quotation|'''WHAT YOU NEED TO DO''':  
 
* Read the two equivalent definitions of index (as number of left cosets, and number of right cosets)
 
* Read the two equivalent definitions of index (as number of left cosets, and number of right cosets)
* Peruse through the rest of the page's content.}}
+
* Convince yourself of why the two definitions are equivalent.}}
{{#lst:Index of a subgroup|main}}
+
{{#lst:Index of a subgroup|beginner}}
 +
{{quotation|'''WHAT'S MORE''': Some further facts about index of a subgroup, some of which you'll see soon in the tour, and others, that'll be seen much later. Ignore those that use unfamiliar terminology.}}
 
{{guided tour-bottom|beginners|Introduction three|Lagrange's theorem|Left and right coset spaces are naturally isomorphic}}
 
{{guided tour-bottom|beginners|Introduction three|Lagrange's theorem|Left and right coset spaces are naturally isomorphic}}

Revision as of 16:18, 1 July 2008

This article adapts material from the main article: index of a subgroup

This page is part of the Groupprops Guided tour for beginners (Jump to beginning of tour)
PREVIOUS: Left and right coset spaces are naturally isomorphic |UP: Introduction three (beginners) | NEXT: Lagrange's theorem
WHAT YOU NEED TO DO:
  • Read the two equivalent definitions of index (as number of left cosets, and number of right cosets)
  • Convince yourself of why the two definitions are equivalent.

Definition

Symbol-free definition

The index of a subgroup in a group is the following equivalent things:

  1. The number of left cosets of the subgroup
  2. The number of right cosets of the subgroup

The collection of left cosets is sometimes termed the coset space, so in this language, the index of a subgroup is the cardinality of its coset space.

Definition with symbols

Given a subgroup H of a group G, the index of H in G, denoted [G:H], is defined in the following ways:

  1. It is the number of left cosets of H in G, i.e. the number of sets of the form xH.
  2. It is the number of right cosets of H in G, i.e. the number of sets of the form Hx.

The collection of left cosets of H in G is sometimes termed the coset space, and is denoted G/H. With this notation, the index of H in G, is the cardinality \left|G/H\right|.

Equivalence of definitions

The equivalence of definitions follows from the fact that there is a natural bijection between the collection of left cosets of a subgroup, and the collection of its right cosets, given by the map g \mapsto g^{-1}

Further information: Left and right coset spaces are naturally isomorphic

WHAT'S MORE: Some further facts about index of a subgroup, some of which you'll see soon in the tour, and others, that'll be seen much later. Ignore those that use unfamiliar terminology.
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: Left and right coset spaces are naturally isomorphic | UP: Introduction three (beginners) | NEXT: Lagrange's theorem