let it be of g for b ejtmulhol/math302/notes/chap18-cosets.pdfcont k 123 e c1237 132 e 53 left...

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Chapter 18 Cosets Let It be a subgroup of G For a b E G define a H b a b E H it is an equivalence relation on 6 reflexive symmetric transitive For ac G the equivalence class a is a be G 1 arab BEG I we call alt ah heh the left coset of It containing a similarly Ha ha halt is a right coset These are the equivalence classes of a rub ab e H Examples Sz E CI 2 I3 23 Cl 237 I 32 It L 23 E CZ3 Coset It Find the right corsets and a set of representatives

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Page 1: Let It be of G For b Ejtmulhol/math302/notes/Chap18-Cosets.pdfcont K 123 E C1237 132 E 53 Left cosets of K K 21,6 0,1 2,3 13 14,15 under t additionmodulo 16 H L47 0,4 8,12 Left Cosets

Chapter 18 Cosets

Let It be a subgroup of G For a b E G define

a H b a b E H

it is an equivalence relation on 6reflexivesymmetric

transitive

For a c G the equivalence class a is

a beG 1 arabBEG I

we call alt ah heh the left coset of It containing a

similarly Ha ha halt is a rightcoset These are the equivalenceclasses of

a rub ab e H

ExamplesSz E CI 2 I 3 23 Cl 237 I 32

It L 23 E CZ3

CosetIt

Findthe rightcorsets and a set of representatives

Page 2: Let It be of G For b Ejtmulhol/math302/notes/Chap18-Cosets.pdfcont K 123 E C1237 132 E 53 Left cosets of K K 21,6 0,1 2,3 13 14,15 under t additionmodulo 16 H L47 0,4 8,12 Left Cosets

contK 123 E C1237 132 E 53Leftcosets of K

K

21,6 0,1 2,3 13 14,15 under t additionmodulo 16

H L47 0,4 8,12LeftCosets of H in 2,6

Proof i it is an equivalence relahai and a alt are the equivalenceclasses Therefore Lemma17.1 implies Ca Cd Cc Part b isa special case of d All that remains to prove is e and f

e

Page 3: Let It be of G For b Ejtmulhol/math302/notes/Chap18-Cosets.pdfcont K 123 E C1237 132 E 53 Left cosets of K K 21,6 0,1 2,3 13 14,15 under t additionmodulo 16 H L47 0,4 8,12 Left Cosets

Lagranges Theorem

Proof

D

It follows that

of distinct leftcosets HI distinctrightcosets

This number is called the indene of It in G

G H IGYIHI

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