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October 4th, 2007, 07:58 PM  #1 
Newbie Joined: Sep 2007 Posts: 9 Thanks: 0  union of disjoint cosets
Let G be a group, B a subgroup of G, and H a subgroup of B. Prove that each coset gB of B in G can be written as a union of disjoint cosets ciH of H in G. ( the i is supposed to be subscript)

October 6th, 2007, 12:13 AM  #2 
Member Joined: Mar 2007 Posts: 57 Thanks: 0 
Cosets are disjoint and form a partition. The cosets of H have an order same as H, and partitions B (and G), and the cosets of B have an order same as B, and partitions G. The rest shouldn't be too hard..........

October 8th, 2007, 09:34 AM  #3 
Newbie Joined: Sep 2007 Posts: 9 Thanks: 0 
i don't get it.

October 9th, 2007, 08:29 PM  #4 
Member Joined: Mar 2007 Posts: 57 Thanks: 0 
Given H is a subgroup of B you know that the cosets of H in B must partition B, and also cosets are disjoint. Therefore the coset eB (which is just B) is a union of disjoint cosets of H, where e is the identity in G. Now you can apply that reasoning to any coset gB of B in G, by thinking what happens when you operate g on (the elements of) each of the cosets of H in the disjoint union for eB, as the coset gB consists of elements of g operated on the elements of the coset eB.


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