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Cosets / Partitions

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  1. Dec 17, 2014 #1
    Hi,
    I am doing self study of Abstract Algebra from Pinter.
    My doubt is regarding Chap 13 Counting Cosets:
    A coset contains all products of the form "ah" where a belongs to G and h belongs to H where H is a subgroup of G. So each coset should contain the number of elements in H. Now the number of cosets should be the number of elements in G since each element of G is used to construct a coset. So the number of elements in all cosets should be number of elements in G*Number of elements in H - But the family of cosets is a partition of G and should have the same number of elements of G..... there is definitely something wrong in the second-third line of this argument... but I am not able to pin it down
     
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  3. Dec 17, 2014 #2

    Stephen Tashi

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    Do two different elements of G necessarily construct two different cosets ?

    For example, suppose the two elements of [itex] G [/itex] are [itex] h1, h2 [/itex] and that [itex] h_1 \in H [/itex] and [itex] h_2 \in H [/itex].
     
  4. Dec 17, 2014 #3
    Thanks - The coset of h1 would be the set of elements h1*h and that of h2 would be h2*h for all h belonging to H - Not sure why they should not be different.
    I am not able to understand why the number of cosets should not be equal to the number of elements in the set G. Am sure I'm missing something very basic.
     
  5. Dec 17, 2014 #4

    Stephen Tashi

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    If [itex] h \in H [/itex] then the coset [itex] hH = H [/itex]. If you multiply an element of a subgroup by another element in the subgroup then the product is in the subgroup.
     
  6. Dec 17, 2014 #5
    Understood - Thanks a ton....
     
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