A
is group, and B
is subgroup of A
, and is an element of A
, then
aB = {ab : b an element of B } is left coset of B in A,The left coset of
B
in A
is subset of A
of form aB
for some a
(element of A
). In aB
(left coset), a
is representative of coset.
and
Ba = {ba : b an element of B } is right coset of B in A.The right coset of
B
in A
is subset of A
of form Ba
for some a
(element of A
). In right coset Ba
, element a
is referred to as representative of coset.
The map aB -> (aB)' = Ba'
map defines bijection between left cosets and B
‘s right cosets, so total of left cosets is equivalent to total of right cosets. The common value is called index of B
in A
.
Left cosets and right cosets are always the same in case of abelian groupings. Notation used switches to a+B
or B+a
if group operation is written additively.
Definition using Equivalence Classes :
Some authors define the left cosets of B
in A
as equivalence classes given by x ~ y
under equivalence relationship on A
if and only if x'y
subset of B
is given. Relation can also be described by x ~ y
if and only if xb = y
is described in B
for certain b
. It can be seen that given relation is simply an equivalence relationship and that two concepts are identical. Consequently, two left B
-in-A
cosets are either equivalent or disjoint. So, every element of A
belongs to single left coset and so left cosets form partition of A
. Similar claims for right cosets are also valid.
Double Cosets :
If A
is group, B
and C
are subgroups of A
, then in A
double coset of B
and C
are sets of BaC
= {bac
: b
an element of B
, c
an element of B
}. These are left cosets of C
and right cosets of B
, respectively, if B
=1 and C
=1.
Notation :
Suppose A
is group and B
and C
.are subgroups of A
.
-
denotes set of left cosetsof B
inA
. -
denotes set of right cosets of B
inA
. -
denotes set of double cosets of B
andC
inA
.
- In computational group theory, cosets are essential.
- Cosets play key role in the theorem for Lagrange.
- The Thistlethwaite’s algorithm used to solve Rubik’s Cube is highly based on cosets.
- linear error-correction in obtained decoded data is done using cosets.
- They are used to construct Vitali sets, kind of non-measurable package.
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