I spent like 2 hours staring at this **** in the exam. I spent much time finding numerical values to check it was correct because it sounded so unlikely... sadly my impressive arithmetic got no marks.
It was certainly challenging! It isn't too long a question really, but the trouble is figuring out how to start. I'm not sure how difficult/long the BMO questions are usually because this is the first one of these I've done. I just did STEP in year 12 and didn't look at BMO.
I remember doing that question, it was nice. I did miss that I can directly conclude 2pโuโv=2kn2 though and ended up doing a slightly longer method of looking at prime factorisation.
I tried to find the official solution after I uploaded mine, but couldn't. One I came across used the prime factorisation method and this was quite a long way of doing it. In my method, I used two different expressions for the RHS and one showed that the expression was even, and the conclusion followed quickly from that. I haven't seen anyone else use that method.
Can someone please recommend books to learn the Numbers and Sets and Differential Equations module for Part IA?
Looking at the syllabus for each of them, they are probably the courses you will least need a book for. Differential equations especially looks really non-rigorous (you'll need some analysis to do differential equations properly). You might benefit slightly from getting a book for numbers and sets like "Naive Set Theory", but it shouldn't be necessary.
Giโ:iโ{1,...,n},G1โร...รGnโโ(G1โร...Giโ)ร(Gi+1โร...)...(...รGnโ) i.e. we can bracket it any way and the group formed is isomorphic to the standard external product of these n groups. But to me this just seems like removing/adding extra brackets from the elements and is straight forwards to prove. so is this a mindless exercise or is there some relevance to other areas? The only thing I can think of is writing elements of R3 as (a,b):aโC,bโR or something
Giโ:iโ{1,...,n},G1โร...รGnโโ(G1โร...Giโ)ร(Gi+1โร...)...(...รGnโ) i.e. we can bracket it any way and the group formed is isomorphic to the standard external product of these n groups. But to me this just seems like removing/adding extra brackets from the elements and is straight forwards to prove. so is this a mindless exercise or is there some relevance to other areas? The only thing I can think of is writing elements of R3 as (a,b):aโC,bโR or something
It should have the same properties as the multiplication of two groups, which is easier to work with. The LHS consist of various n-tuples of elements - one element taken from each group and then the multiplication is done 'component wise' if I remember my group theory. Why would it have to be in R3?
It should have the same properties as the multiplication of two groups, which is easier to work with. The LHS consist of various n-tuples of elements - one element taken from each group and then the multiplication is done 'component wise' if I remember my group theory. Why would it have to be in R3?
yeah thats just the definition of direct products, but my question was what relevance does this exercise have? So as an example I gave the example of rewriting R3 as effectively '2-tuples' as ((a,b),c) instead of the usual (a,b,c). But still i don't really see the significance of this result overall.
yeah thats just the definition of direct products, but my question was what relevance does this exercise have? So as an example I gave the example of rewriting R3 as effectively '2-tuples' as ((a,b),c) instead of the usual (a,b,c). But still i don't really see the significance of this result overall.
If you're looking for an example where this might be useful or relevant then I can't really think of one. It's still a result that is good to know though.
yeah thats just the definition of direct products, but my question was what relevance does this exercise have? So as an example I gave the example of rewriting R3 as effectively '2-tuples' as ((a,b),c) instead of the usual (a,b,c). But still i don't really see the significance of this result overall.
I can think of two lines of answer to this question. The first is purely didactic: you are given the definition of a product of groups and showing that the re-bracketing property hold shows that you really have understood the definitions. Learning rigour, it's good for the soul!
The second is to think about where such a re-bracketing property is not true. In the category of setsAร(BรC)๎ =(AรB)รC, for example. So, why is it true for groups? What has happened? Why has replacing equals with isomorphism made such a difference? Can you come up with non-trivial algebraic examples where it fails even for isomorphism?
I can think of two lines of answer to this question. The first is purely didactic: you are given the definition of a product of groups and showing that the re-bracketing property hold shows that you really have understood the definitions. Learning rigour, it's good for the soul!
The second is to think about where such a re-bracketing property is not true. In the category of setsAร(BรC)๎ =(AรB)รC, for example. So, why is it true for groups? What has happened? Why has replacing equals with isomorphism made such a difference? Can you come up with non-trivial algebraic examples where it fails even for isomorphism?
Well in the case A,B,C are groups, algebraically its clear to show it holds by looking at the mapping ฯ((a,b),c))=(a,(b,c)) and showing its one to one and onto. If the product of n-tuples is done element wise, ฯ((a,b),c)((d,e),f))=ฯ((ad,be),cf)=(ad,(be,cf))=(a,(b,c))(d,(e,f))=ฯ((a,b),c))ฯ((d,e),f)) so the two are isomorphic. Since the composition of two isomorphisms is an isomorphism we can apply this multiple times to any number of groups to bracket it how u want. So I think the isomorphism makes the difference as it distinguishes between the elements of each set. e.g. if (a,(b,c))=((a,b),c) then a=(a,b),(b,c)=c which clearly can't hold, as the elements on either side of the equals are in different sets. Isomorphisms are usually described as 'structure preserving mappings' informally so if we had a plain old set (i.e. with no structure) we'd just be talking about bijection id imagine so if that is the case Aร(BรC)โ(AรB)รC still makes sense if we're dealing with sets. When you say "Can you come up with non-trivial algebraic examples where it fails even for isomorphism?" do you mean in reference to the bracketing of these algebraic structures being isomorphic (or not in this case)?
When you say "Can you come up with non-trivial algebraic examples where it fails even for isomorphism?" do you mean in reference to the bracketing of these algebraic structures being isomorphic (or not in this case)?
hmm im not sure what example to be looking for, cus is the definition of an isomorphism the same for all algebraic structures? Cus I've heard of ring isomorphisms but not much about them. Does this counterexample need to hold for all the isomorphisms different to the usual ฯ(xy)=ฯ(x)ฯ(y) one? (assuming they are different).
@EnglishMuon Here's another question that you might like... (it's quite straightforward but part (b) draws on knowledge of other groups)
Consider the group G=(Z2โ,+) (a) Construct the composition table for the external product GรG. (b) Can you think of a group that you may have already studied that is isomorphic to GรG?
@EnglishMuon Here's another question that you might like... (it's quite straightforward but part (b) draws on knowledge of other groups)
Consider the group G=(Z2โ,+) (a) Construct the composition table for the external product GรG. (b) Can you think of a group that you may have already studied that is isomorphic to GรG?
Assuming the addition is mod2: Z2โ={0,1} so Z2โรZ2โ={(0,0),(1,0),(0,1),(1,1)}.
This is isomorphic to the symmetries of the rectangle,
D4โ={xiyj:i,j=0,1ย xy=yโ1x(=yx)} by the isomorphism defined by ฯ(xiyj)=(i,j) . e.g. as xiyjxayb=xi+ayj+b as abelian (hopefully)