Distinct left(right) cosets

  • Thread starter ivyhawk
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  • #1
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Main Question or Discussion Point

G group, H subgroup of G.
Suppose aH and bH are distinct leftcosets then Ha and Hb must be distinct right cosets?


My humble thoughts:
the left coset aH consists of a times everything in H;
Ha consists of everything in H times a.
Then this argument above is true?
 

Answers and Replies

  • #2
Fredrik
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That's not an argument, it's just the definition of the cosets. :smile:
 
  • #3
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That's not an argument, it's just the definition of the cosets. :smile:
My doubt is whether there is a counter-example such that when aH and bH are distinct left cosets, Ha and Hb are not distinct right cosets, because this statement looks suspicious.
 
  • #4
Fredrik
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When H is a normal subgroup, it's very easy to prove that the statement is true, but I don't see why it must be true when H isn't a normal subgroup. So if you're looking for counterexamples, start by thinking of examples of subgroups that aren't normal.
 

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