Left coset of a subgroup: Difference between revisions

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Definition

Definition with symbols

Let $H$ be a subgroup of a group $G$ . Then, a left coset of $H$ is a nonempty set $S\subset G$ satisfying the following equivalent properties:

$x^{-1}y$ is in $H$ for any $x$ and $y$ in $S$ , and for any fixed $x\in S$ , the map $y\mapsto x^{-1}y$ is a surjection from $S$ to $H$
There exists an $x$ in $G$ such that $S=xH$ (here $xH$ is the set of all $xh$ with $h\in H$ )
For any $x$ in $S$ , $S=xH$

Any element $x\in S$ is termed a coset representative for $S$ .

Equivalence of definitions

For full proof, refer: Equivalence of definitions of left coset

Facts

Left congruence

The left cosets of a subgroup are pairwise disjoint, and hence form a partition of the group. The relation of being in the same left coset is an equivalence relation on the group, and this equivalence relation is termed the left congruence induced by the subgroup.

Relation with right coset

Every subset that occurs as a left coset of a subgroup also occurs as a right coset. In fact, the left coset $xH$ occurs as the right coset $(xHx^{-1})x$ with $xHx^{-1}$ being the new subgroup.

Natural isomorphism of left cosets with right cosets

There is a natural bijection between the set of left cosets of a subgroup and the set of right cosets of that subgroup. This bijection arises from the natural antiautomorphism of a group defined by the map sending each element to its inverse. Further information: Left and right coset spaces are naturally isomorphic

Numerical facts

Size of each left coset

Let $H$ be a subgroup of $G$ and $x$ be any element of $G$ . Then, the map sending $g$ in $H$ to $xg$ is a bijection from $H$ to $xH$ .

For full proof, refer: Left cosets are in bijection via left multiplication

Number of left cosets

The number of left cosets of a subgroup is termed the index of that subgroup.

Since all left cosets have the same size as the subgroup, we have a formula for the index of the subgroup when the whole group is finite: it is the ratio of the order of the group to the order of the subgroup.

This incidentally also proves Lagrange's theorem -- the order of any subgroup of a finite group divides the order of the whole group.

References

Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, ^{More info}, Page 77 (formal definition, along with right coset)
Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261, ^{More info}, Page 5 (definition introduced in paragraph)
A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613, ^{More info}, Page 10
An Introduction to Abstract Algebra by Derek J. S. Robinson, ISBN 3110175444, ^{More info}, Page 51
Algebra by Serge Lang, ISBN 038795385X, ^{More info}, Page 12
A First Course in Abstract Algebra (6th Edition) by John B. Fraleigh, ISBN 0201763907, ^{More info}, Page 121 (formal definition, along with right coset)
Algebra (Graduate Texts in Mathematics) by Thomas W. Hungerford, ISBN 0387905189, ^{More info}, Page 38 (definition introduced after Theorem 4.2, which is about left congruence and right congruence; introduced along with right coset)
Contemporary Abstract Algeba by Joseph Gallian, ISBN 0618514716, ^{More info}, Page 132
Topics in Algebra by I. N. Herstein, ^{More info}, Page 47, Exercise 5 (definition introduced in exercise)

@@ Line 7: / Line 7: @@
 Let <math>H</math> be a [[subgroup]] of a [[group]] <math>G</math>. Then, a '''left coset''' of <math>H</math> is a nonempty set <math>S \subset G</math> satisfying the following equivalent properties:
-* <math>x^{-1}y</math> is in <math>H</math> for any <math>x</math> and <math>y</math> in <math>S</math>, and for any fixed <math>x \in S</math>, the map <math>y \mapsto x^{-1}y</math> is a surjection from <math>S</math> to <math>H</math>
+# <math>x^{-1}y</math> is in <math>H</math> for any <math>x</math> and <math>y</math> in <math>S</math>, and for any fixed <math>x \in S</math>, the map <math>y \mapsto x^{-1}y</math> is a surjection from <math>S</math> to <math>H</math>
-* There exists an <math>x</math> in <math>G</math> such that <math>S = xH</math> (here <math>xH</math> is the set of all <math>xh</math> with <math>h \in H</math>)
+# There exists an <math>x</math> in <math>G</math> such that <math>S = xH</math> (here <math>xH</math> is the set of all <math>xh</math> with <math>h \in H</math>)
-* For any <math>x</math> in <math>S</math>, <math>S = xH</math>
+# For any <math>x</math> in <math>S</math>, <math>S = xH</math>
+Any element <math>x \in S</math> is termed a ''coset representative'' for <math>S</math>.
 ===Equivalence of definitions===
@@ Line 22: / Line 24: @@
 Every subset that occurs as a left coset of a subgroup also occurs as a [[right coset]]. In fact, the left coset <math>xH</math> occurs as the right coset <math>(xHx^{-1})x</math> with <math>xHx^{-1}</math> being the new subgroup.
+===Natural isomorphism of left cosets with right cosets===
+There is a natural bijection between the set of left cosets of a subgroup and the set of right cosets of that subgroup. This bijection arises from the natural antiautomorphism of a group defined by the map sending each element to its inverse. {{further|[[Left and right coset spaces are naturally isomorphic]]}}
 ==Numerical facts==
@@ Line 39: / Line 45: @@
 This incidentally also proves [[Lagrange's theorem]] -- the order of any subgroup of a finite group divides the order of the whole group.
-==Natural isomorphism of left cosets with right cosets==
+<section end=main/>
+==References==
-There is a natural bijection between the set of left cosets of a subgroup and the set of right cosets of that subgroup. This bijection arises from the natural antiautomorphism of a group defined by the map sending each element to its inverse. {{further|[[Left and right coset spaces are naturally isomorphic]]}}
+* {{booklink-defined|DummitFoote}}, Page 77 (formal definition, along with [[right coset]])
-<section end=main/>
+* {{booklink-defined|AlperinBell}}, Page 5 (definition introduced in paragraph)
+* {{booklink-defined|RobinsonGT}}, Page 10
+* {{booklink-defined|RobinsonAA}}, Page 51
+* {{booklink-defined|Lang}}, Page 12
+* {{booklink-defined|Fraleigh}}, Page 121 (formal definition, along with [[right coset]])
+* {{booklink-defined|Hungerford}}, Page 38 (definition introduced after Theorem 4.2, which is about [[left congruence]] and [[right congruence]]; introduced along with [[right coset]])
+* {{booklink-defined|Gallian}}, Page 132
+* {{booklink-defined|Herstein}}, Page 47, Exercise 5 (definition introduced in exercise)