Mathematics, 01.11.2019 03:31 sbhunsaker9722
Let h and k be subgroups of a group g, and let g be an element of g. the set 
is called a double coset. do the double cosets partition g?
Answers: 3
. We say that 
if, and only if, 
, or, equivalently, if 
, for some 
.
for all 
.) Note that we can write 
, where 
is the identity element, so 
and then 
. Therefore, 
.) If 
for some 
and 
. Multiplying by the inverses of h and k we get that 
and is known that 
and 
. This means that 
or, equivalently, 
, then 
.) If 
and 
such that 
and 
. Then, 
where 
and 
. Consequently, 
.
" is an equivalence over G, we use the fact that the different equivalence classes partition G. Since the equivalence classes are defined by ![[x]=\{y\in G\colon x\sim y\} =\{y\in G \colon y=hgk\ \text{for some } h\in H, k\in K \}=HxK](/tpl/images/0355/1781/f529f.png)
, then we're done.
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Let h and k be subgroups of a group g, and let g be an element of g. the set [tex]\math hgk = \{x \i...
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