PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (200)
Orphanage
Unclass'd
Unproven (490)
Corrections (6)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: High Entry average rating: Very high
coset (Definition)

Let $ H$ be a subgroup of a group $ G$, and let $ a \in G$. The left coset of $ a$ with respect to $ H$ in $ G$ is defined to be the set

$\displaystyle aH := \{ ah \mid h \in H \}. $
The right coset of $ a$ with respect to $ H$ in $ G$ is defined to be the set
$\displaystyle Ha := \{ ha \mid h \in H \}. $
Two left cosets $ aH$ and $ bH$ of $ H$ in $ G$ are either identical or disjoint. Indeed, if $ c \in aH \cap bH$, then $ c = ah_1$ and $ c = bh_2$ for some $ h_1,h_2 \in H$, whence $ b^{-1} a = h_2 h_1^{-1} \in H$. But then, given any $ ah \in aH$, we have $ ah = (bb^{-1})ah = b(b^{-1}a) h \in bH$, so $ aH \subset bH$, and similarly $ bH \subset aH$. Therefore $ aH = bH$.

Similarly, any two right cosets $ Ha$ and $ Hb$ of $ H$ in $ G$ are either identical or disjoint. Accordingly, the collection of left cosets (or right cosets) partitions the group $ G$; the corresponding equivalence relation for left cosets can be described succintly by the relation $ a \sim b$ if $ a^{-1} b \in H$, and for right cosets by $ a \sim b$ if $ ab^{-1} \in H$.

The index of $ H$ in $ G$, denoted $ [G:H]$, is the cardinality of the set $ G/H$ of left cosets of $ H$ in $ G$.



"coset" is owned by djao. [ full author list (2) ]
(view preamble)

View style:

Also defines:  index, left coset, right coset
Log in to rate this entry.
(view current ratings)

Cross-references: cardinality, relation, equivalence relation, partitions, collection, disjoint, group, subgroup
There are 57 references to this entry.

This is version 5 of coset, born on 2002-01-05, modified 2002-11-04.
Object id is 1306, canonical name is Coset.
Accessed 20222 times total.

Classification:
AMS MSC: 20A05 (Group theory and generalizations :: Foundations :: Axiomatics and elementary properties)
Pending Errata and Addenda
None.
[ View all 1 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)