Tour:Left cosets are in bijection via left multiplication
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This article adapts material from the main article: left cosets are in bijection via left multiplication
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The different left cosets of a subgroup are in bijection with each other, under the action of the group via left multiplication. This means that all left cosets of a subgroup are of the same size.
PREREQUISITES: Definition of group, subgroup and coset, invertible implies cancellative, and Tour:Mind's eye test two (beginners)#Left and right multiplication maps
WHAT YOU NEED TO DO: Read and understand the statement and proof below. If you find it hard, refer back to the suggested prerequisites pages.
Statement
Statement with symbols
Let be a subgroup of a group
and let
and
be two left cosets of
. Then, there is a bijection between
and
as subsets of
, given by the left multiplication by
.
Facts used
- Invertible implies cancellative in monoid: In particular, we can cancel elements in a group: if
, then
.
Proof
Given: A group , a subgroup
, and two left cosets
,
of
To prove: Left multiplication by establishes a bijection between
and
.
Proof: We prove that left multiplication by sends
to
, is surjective, and is injective.
- Well-defined as a map from
to
: First, note that if
then
. Thus, any element in
gets mapped to an element in
.
- Surjective: Every element of the form
with
in
arises as
, hence, it arises as the image of left multiplication by
. Thus, the map from
to
is surjective.
- Injective: Given two distinct elements
, the elements
and
are also distinct, because by fact (1), if they were equal, then canceling
from both sides would give
. Thus, left multiplication by
sends distinct elements to distinct elements, so the map is injective.
Thus, left multiplication by is a bijection from
to
.
Sidenote
In general, there is no natural bijection between two left cosets -- the bijection depends on a choice of element in both cosets (the elements and
in the above description).
This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)
PREVIOUS: Left cosets partition a group| UP: Introduction three (beginners)| NEXT: Right coset of a subgroup
General instructions for the tour | Pedagogical notes for the tour | Pedagogical notes for this part