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A Cayley diagram problem 
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#1
Feb711, 05:10 PM

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1. The problem statement, all variables and given/known data
Given the four Cayley diagrams in the attachment, one needs to determine which of the groups represented by them are isomorphic. 3. The attempt at a solution Well, after inspecting all of them, I concluded none are isomorphic. For the first two, after constructing the tables, one can find a product where the identity elements don't coincide. The same with the third one and the first two. Now, the fourth diagram, unless I'm mistaken, doesn't even have 6 elements! Since the full line represents multiplication with the generator b and b^1 = b. So it can't be isomorphic to any of the other ones. I'm not really sure about my results. If you want, I can provide more details. Thanks in advance for a glance. 


#2
Feb711, 05:29 PM

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For example, to elaborate on the last diagram  let any point be the identity element e. Then, starting clockwise for example, we generate the elements b, b^2 = e (in my book the convention is used that a line without an arrowhead means that b^2 = e), b^3 = b, b^4 = e, b^5 = b, and we end at b^6 = e. Hence, this group has elements b and e.
Edit: the example of the first diagram can already be found in the book (see attachment), and the second is the same, only ab and ab^2 are interchanged, unless I'm wrong. 


#3
Feb711, 06:00 PM

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Hmm, the answer is certainly not correct. There are some Cayley graphs which come from thesame groups!!
In fact, it would be easier if for every graph, you list the group where the graph comes from. For example, the last diagram: clearly, if we choose an element b, then the other elements are of the form b^{n}. Thus this clearly is a cyclic group. In fact, the last graph is the Cayley graph of the group [tex]\mathbb{Z}_6[/tex]. (where I'm confused is when you say that the last graph doesn't have 6 elements. Why do you conclude that? A graph has as much elements as it's vertices)... Try to write down explicitely the groups (or even try working out their Cayley multiplication table). Maybe a little hint: there are only two groups of order 6: [tex]\mathbb{Z}_6[/tex] and [tex]S_3[/tex]. So this could help you... 


#4
Feb711, 06:04 PM

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A Cayley diagram problem
Maybe start with this: which of the Cayley diagrams come from abelian groups?? That's not hard to find out I think...



#5
Feb811, 12:58 AM

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Ah, of course, I just realized that points on a Cayley diagram stand for distinct elements  however, what confused me was the convention in the book which says that "arrows with no arrowheads mean that b^2 = e, where these arrows mean multiplying from the right with b". There's something confusing about that. If the group represented by the last diagram is Cyclic, then why don't the lines connecting the points have arrowheads?



#6
Feb811, 02:17 AM

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OK, about the first two diagrams.
As I mentioned, the second one is the same as the first, except ab and ab^2 are interchanged. I wrote down the group table, but even without it, if you follow the diagram, it is obvious that for this group we have ab^2 * ab = e, whereas in the first one we have ab^2 * ab = b^2. Since if two groups G and H are isomorphic, then for xy = e in G, we have f(xy) = f(x)f(y) = e in H. For example, an obvious bijection for the upper examples would be the identity! The groups even consist of the same elements. But the structure is not preserved  the products of the images of x and y do not equal the inverse in H (i.e. the second Cayley diagram in our case). 


#7
Feb811, 07:48 AM

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Hmm, I really don't know why the author doesn't write arrowheads on the last diagram...
So in order to prove this, you have to write down every possible bijection and check if it is an homomorphism. However, there are 6! possible bijections, so you would not want to do that. Instead, try to find a qualitative difference between the two diagrams. For example, the first diagram yields a nonabelian group and the second is an abelian group. This would suffice to show that the two groups are not isomorphic... 


#8
Feb811, 07:58 AM

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. Edit: I was trying to prove or disprove an isomorphism relation by merely looking at the multiplication tables of the groups, which was actually a bit misleading. Actually, this only is of use when the tables "look the same" (I'm being very informal now, but I think you know what I mean, i.e. if we identify every element of a table for G with another element of H in a onetoone manner, then the product of any two elements in G must be on the same place as the product of the elements in H which correspond to our starting elements in G). 


#9
Feb811, 08:08 AM

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#10
Feb811, 08:11 AM

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#11
Feb811, 09:30 AM

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OK, let's first resolve the last two groups.
The last group is a cyclic group (generated by b) with elements {e, b, b^2, b^3, b^4, b^5}, of course after we added the arrowheads to the diagram, since without them it doesn't seem to make any sense. This group is obviously abelian. Now, let's look at the diagram before the last one. So, we get a group consisting of {e, b, ba, bab, baba, babab}. From the diagram we can conclude that a = a^1 and b = b^1. Actually, in this diagram, the notation without arrowheads makes sense. This group is not abelian, since for example ba*bab = babab, and bab*ba = b. Hence, the last two groups are not isomorphic. Is this correct? 


#12
Feb811, 09:46 AM

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Further on, the first group is not abelian, since ba = ab^2 and ab = ab. But the second one is (I wrote down its table and it is symmetric). Hence (I'll refer to the diagrams, i.e. the groups as 1, 2, 3 and 4 respectively according to the attachment from now on) 4 is not isomorphic to 3, neither to 1.



#13
Feb811, 10:31 AM

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Yes, that is correct. 4 is not isomorphic to either 1 or 3. But I claim that it is isomorphic to 2. And sadly there is no other way to prove this than to construct an explicit isomorphism...
Thesame technique shows that 3 is not isomorphic to either 2 or 4. But it probably is isomorphic to 1. In fact, 1 and 3 both come from the group S_{3}... 


#14
Feb811, 12:42 PM

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I have both tables. First of all, the element having itself as an inverse must be matched to the element with the same property. So I obtain b^3 > a. Now I look where b^3 occurs and obtain b^5 > b^2 and b^4 > ab. But when I proceed with the other 3 elements, I always find a multiplication entry which is not correct. I think I may be missing something here. I checked my tables, they seem to be OK. 


#15
Feb811, 12:46 PM

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Actually, the relations I wrote above (except for b^3) won't work either.
If the point of this is to find a combination that works, then I won't do it. Just verify if my thought is correct. Edit: and btw, did you mean that 1 and 3 both come from S6? Or I'm missing something here? It is, btw, obvious that 4 is isomorphic to the addition of integers modulo 5 (in fact, it seems to me I've seen some result before and it seems quite expectable that every finite cyclic group opf order n is isomorphic to a certain group of integers under addition modulo n1, probably the same holds for infinite cyclic groups and the set of integers with standard addition), but it seems less obvious that 1 and 3 are isomorphic to S6...or it's trivial  these are simply groups with the same number of elements as S6 and the entries in the rows are simply permutations of 6 elements, nothing more... 


#16
Feb811, 12:56 PM

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I had something different: I did b > ab. But your choice should work to. But the sad thing about these kinds of exercises that it actually comes down to finding the right isomorphism. A quicker proof is possible when you solved 13.F, that exercise states that there are exactly 2 groups of order 6. That implies immediately that an isomorphism exists and then you won't have to construct it explicitely... 


#17
Feb811, 01:02 PM

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#18
Feb811, 01:04 PM

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