Why is H * a not equal to a * H Please solve a question quickly

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Why is H * a not equal to a * H Please solve a question quickly
Definition:
*- Let (G,*) be a group and let (H,*) be a subgroup of (G,*) we
define a set H * a by
H*a = {h * a, he H},Va E G is a set of a right coset of H in G
%3D
Similarly, we define a left coset of H in G by
a * H = {a * h, he H},Va E G
{a * h, h e H}, Va E G
**- In general H * a # a * H
***- The quotient set °/ =.mod H
%3D
E. mod H =H * a, a E G}is the set
of all equivalence classes ( right or left ) of H in G
G
****- We shall denote / =. mod H
by C/H
i.el G/H = {H * a, a E G} or {a * H, a E G}
%3D
proposition:
Transcribed Image Text:Definition: *- Let (G,*) be a group and let (H,*) be a subgroup of (G,*) we define a set H * a by H*a = {h * a, he H},Va E G is a set of a right coset of H in G %3D Similarly, we define a left coset of H in G by a * H = {a * h, he H},Va E G {a * h, h e H}, Va E G **- In general H * a # a * H ***- The quotient set °/ =.mod H %3D E. mod H =H * a, a E G}is the set of all equivalence classes ( right or left ) of H in G G ****- We shall denote / =. mod H by C/H i.el G/H = {H * a, a E G} or {a * H, a E G} %3D proposition:
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