  # Let H and K be subgroups of a finite group G. Show that|HK|HK=|HОКIwhere HK (hk hE H, k E K}. (HK is not assumed to be a subgroup of G):

Question help_outlineImage TranscriptioncloseLet H and K be subgroups of a finite group G. Show that |HK |HK= |HОКI where HK (hk hE H, k E K}. (HK is not assumed to be a subgroup of G) : fullscreen
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Step 1

let D = HK then D is a subgroup of k and there exist a decomposition of k into disjoint right cosets of D in k and

Step 2

This impl... help_outlineImage Transcriptionclose|k=|Dk; U Dk,U..Dk, = Dk+|Dk+..D ID+|D|+..D (t times) ..(2) t = Again Hk H Dk; i1 i1 =UHk, DCH) i1 =Hk UHk,U...U Hk Now no two of Hk,,Hk,.. .Hk, can be equal as if Hk, = Hk, for some i,j then, kkHkkEHNk = D kk,ED Dk, Dk, fullscreen

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