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

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In summary, the question is whether a and b must be in a subgroup if the product ab is in the subgroup. It is true that if a and b are in the subgroup, then ab is also in the subgroup. However, the reverse may not necessarily be true. The identity element e is always in every subgroup, but not all elements of the group may be in every subgroup. Additionally, (ab)-1 is always equal to b-1a-1, which can be proven by multiplying b^(-1)a^(-1) and ab.
  • #1
Ryker
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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.
 
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  • #2
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
Ah, I see, so it only works one way then? Thanks for the quick response.
 
  • #4
Oh, sorry, one more thing, (ab)-1 is always b-1a-1, though, right?
 
  • #5
Ryker said:
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
Yeah, that's what I was thinking :smile:
 

1. Is it always true that if ab is in a subgroup, then a and b must also be in the subgroup?

Yes, this is a fundamental property of subgroups. If the product of two elements is in a subgroup, then each individual element must also be in the subgroup.

2. Can you provide an example of a subgroup where ab is in the subgroup but a and b are not?

No, this is not possible. As stated before, if ab is in a subgroup, then a and b must also be in the subgroup.

3. Does this property also apply to other algebraic structures, such as rings or fields?

Yes, this property is true for any algebraic structure that has a subgroup operation. It is a fundamental concept in abstract algebra.

4. Can this property be used to determine if a set is a subgroup?

No, this property alone is not enough to determine if a set is a subgroup. Other conditions, such as closure under the subgroup operation and the existence of an identity element, must also be satisfied.

5. Are there any exceptions to this property?

No, this property is always true for subgroups. However, it may not hold for other types of subsets in a mathematical structure.

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