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

[Ryker]

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.

 

[Dick]

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.

 

[Ryker]

Ah, I see, so it only works one way then? Thanks for the quick response.

 

[Ryker]

Oh, sorry, one more thing, (ab)^-1 is always b^-1a^-1, though, right?

 

[Dick]

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?

 

[Ryker]

Yeah, that's what I was thinking