Difference between revisions of "Coset"
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== See also == | == See also == | ||
− | * [[Lagrange's | + | * [[Lagrange's Theorem]] |
* [[Quotient group]] | * [[Quotient group]] | ||
* [[Orbit]] | * [[Orbit]] | ||
[[Category:Group theory]] | [[Category:Group theory]] |
Latest revision as of 01:28, 3 September 2008
A coset is a subset of a group.
Specifically, let be a group, and let
be a subgroup of
. The left cosets modulo
are the subsets of
of the form
, for
. Note that for any coset
, the mapping
is a bijection from
to
. Hence for any
,
.
The image of a left coset under the mapping
is the right coset
. This mapping induces a bijection from the set of left cosets of
to the set of right cosets of
.
The cardinality of the set of left cosets of is called the index of
with respect to
; it is denoted
. This is also the cardinality of the set of right cosets of
.
Proposition. The relations ,
are equivalence relations.
Proof. We prove that the first relation is an equivalence relation; the second then follows by passing to the opposite law on .
We abbreviate as
. For any
, note that
, so
. If
, then
, so
implies
. Finally, if
and
, then
; hence
and
together imply
. Hence
is an equivalence relation.
Cosets and compatible relations
We call a relation left compatible with the group structure of
if
implies
, for all
. Similarly, we say
is right compatible with the group structure of
if
implies
. Note that
is compatible with the group law on
if and only if it is both left- and right-compatible with the structure.
Theorem. An equivalence relation on a group
is left (resp. right) compatible with
if and only if it is of the form
(resp.
), for some subgroup
of
. In this case,
is the equivalence class of
, the identity, and the equivalence classes are the left (resp. right) cosets of
.
Proof. We will consider only the case for left compatible with
; the other case follows from symmetry.
Let be the equivalence class of
. Note that
if and only if
, which is true if and only if
. It thus remains to show that
is a subgroup of
.
To this end, we note that evidently ; also, if
, then
, so
. Finally, if
are in
, then
. Thus
is a subgroup of
.
Conversely, suppose is a subgroup of
, and define
as
. We have proven that
is an equivalence relation; evidently
if and only if
. Now, if
, then
, so
is left-compatible with the group structure of
.
Now, if and only if
;. Hence the set of
equivalent to
(mod
) is the set
. Thus the equivalence classes of
are the left cosets mod
.