If ab is in a subgroup, are a and b neccessarily in the subgroup?

  • Thread starter Ryker
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  • #1
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Homework Statement


I was just wondering if the product ab is in a subgroup, are a and b necessarily in the subgroup, as well?

The Attempt at a Solution


I think they are, but how would you prove that? Or is that obvious from closure under multiplication and you don't need to prove it? I know it works the "normal" way, that is if a and b are in the subgroup (or group), then ab is in it, as well, but I'm not sure about the reverse direction.

Thanks in advance for any replies.
 

Answers and Replies

  • #2
Dick
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The identity e is in every subgroup. a*a^(-1)=e for all elements of the group. That doesn't mean a is in every subgroup.
 
  • #3
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Ah, I see, so it only works one way then? Thanks for the quick response.
 
  • #4
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Oh, sorry, one more thing, (ab)-1 is always b-1a-1, though, right?
 
  • #5
Dick
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Oh, sorry, one more thing, (ab)-1 is always b-1a-1, though, right?

Sure. Multiply b^(-1)a^(-1) and ab. What do you get?
 
  • #6
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Yeah, that's what I was thinking :smile:
 

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