Coset in a group
by a subgroup
(from the left)
A set of elements of of the form
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where is some fixed element of
. This coset is also called the left coset by
in
defined by
. Every left coset is determined by any of its elements.
if and only if
. For all
the cosets
and
are either equal or disjoint. Thus,
decomposes into pairwise disjoint left cosets by
; this decomposition is called the left decomposition of
with respect to
. Similarly one defines right cosets (as sets
,
) and also the right decomposition of
with respect to
. These decompositions consist of the same number of cosets (in the infinite case, their cardinalities are equal). This number (cardinality) is called the index of the subgroup
in
. For normal subgroups, the left and right decompositions coincide, and in this case one simply speaks of the decomposition of a group with respect to a normal subgroup.
Comments
See also Normal subgroup.
Coset in a group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Coset_in_a_group&oldid=33616