Proof of One Step Subgroup TestDate: 05/13/2002 at 07:00:04 From: Sarah Hill Subject: algebra subgroup proof Hi, I am stuck with how to do the following proof: Prove that a nonempty subset H of a group G is a subgroup of G if and only if a*b^(-1) is in H for all a, b in H. I am pretty sure that I now have to prove the 4 axioms, that: 1. G is closed under * 2. * is associative 3. the identity e is in G 4. there exists an inverse element b for every a in G. I know the operation * is associative as H has the same operation as G, but i am unsure how to prove the others, especially the inverse and identity ones. Thank you for your help. Sarah Date: 05/13/2002 at 09:06:55 From: Doctor Paul Subject: Re: algebra subgroup proof Hi Sarah, This is a really powerful theorem. It's generally known as the "One Step Subgroup Test." Instead of having to show that all of the group axioms hold, you only need to show this one result and then the group axioms follow. You wrote: >I am pretty sure that I now have to prove the 4 axioms, that: >1. G is closed under * >2. * is associative >3. the identity e is in G >4. there exists an inverse element b for every a in G. This is for proving the reverse direction -- you're assuming the condition holds and then you want to prove that under the assumption that H is a subgroup of G. But the conditions above aren't quite right. We don't want to show (1), (2), and (4) for G. We are told that G is a group so we already know that these hold for G. We want to show them for H. So we want to show: 1. H is closed under * 2. * is associative 3. the identity e (of the group G) is in H 4. there exists an inverse element b for every a in H. The operation * is associative on G (since G is a group) so certainly it will be associative on any subset of G. This shows (2). Now we need to more clearly state our hypothesis: Suppose that G is a group, H is a nonempty subset of G, and that a*b^(-1) is in H for all a, b in H. We want to use these hypotheses to prove (1), (3), and (4). To see (3), suppose that a*b^(-1) is in H for all a, b in H. In other words, whenever we pick two elements a and b from H, we know that the product a*b^(-1) will be in H. The key is to make clever choices for a and b. This will become evident below. Note that we are assuming H is a nonempty subset of G. So we know that H contains at least one element. Let's consider the simplest case -- what if H contained only one element? Then if we were choosing elements a and b from H, we would be forced to choose a = b. But we know that whenever we choose two elements (call them a and b) from H that a*b^(-1) is in H. In the special case where a = b, we have a*b^(-1) = a*a^(-1) = e is in H. This gives the desired result. But we have no reason to think that assuming H contains only one element is an okay thing to do. So we need to deal with the case where H could contain more than one element. So suppose H is a subset of G that contains more than one element. We want to pick elements a and b from H. Here, we are no longer forced to pick a = b. Just pick a = b anyway. There is nothing in the statement of the hypothesis that says we have to pick different elements for a and b. And picking a = b gives the desired result (see the previous paragraph). To see (4) I guess we should consider two cases for completeness. We know from above that H contains the identity element. Case 1: H has only one element. We know from (3) that H contains the identity and we are assuming H contains only one element so it must be the case that H = {e}. Certainly e^(-1) = e so for every element in H (there is only one element, namely e, for which we need to check this condition) there exists another element in H which is the inverse of the first element. Case 2: H has more than one element. We know from (3) that H contains the identity element e. And we are assuming it contains at least one other element. Pick one of these additional elements at random and call it x. We know from Case 1 above that there exists an inverse for e in H. We want to show that there exists an inverse for x in H. To see this, we use our hypothesis. We know that whenever we pick two elements a and b from H that a*b^(-1) is in H. Here, choose a = e, b = x. Certainly e and x are in H. Then e*x^(-1) = x^(-1) is in H. This is what we wanted to show. Now we show (1) -- that H is closed under *. Suppose x, y are in H. We can use the hypothesis to note that x*y^(-1) is in H. But that isn't what we want. We what to conclude is that x*y is in H. Well, if x and y are in H then we know by (4) that y^(-1) is in H as well. Our hypothesis says that whenever we choose two elements a and b from H that the product a*b^(-1) will be in H. Here, choose a = x, b = y^(-1). Then a*b^(-1) = x*y will be in H. This completes the reverse direction. Now we need to show the forward direction -- that if H is a subgroup then a*b^(-1) is in H for all a, b in H. This is obvious isn't it? We are assuming that H is a group. So H is closed under taking inverses and multiplication. I believe this completes the proof. Again, the trick is that you have to make intelligent choices about how you choose the elements a and b from H. I hope this helps. I've tried to make it as clear as possible, even including more explanation than I thought was necessary. I did this because I remember having trouble with this proof the first time I took a course in modern algebra. If something remains unclear, please write back and we'll work through it. Just be sure to let me know where you're having trouble. - Doctor Paul, The Math Forum http://mathforum.org/dr.math/ |
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