matt grime

Let me clarify what your definition of an affine map is: is it one that preserves lines or is L(x) + v for some linear map L? (v=F(0) incidentally) Since you've said that the line is mapped to the other line then since the original line is in C its image must be in F©, agreed? end of proof.

pick two points in F© say F(x), F(y) Then as F is affine the line x+t(y-x) t in[0,1] is... if it helps F(x) = L(x)+v for some linear map L and some fixed vector v, though I presume you know that (since the other definition of affine is that i preserves straight lines, making the question easier).

You just do it: start with 1, where is one sent? to x, say, where is x sent? to y, say, then where is y sent to? repeat until you hit 1, then the first of the cycles is: (1xy...) then pick some element not yet hit, and start again. For instance, to simplify (123)(345)(43)(675) 1 is missed by 675, and the 43, and the 345, but 123 sends it to 2. 2 is missed by 675, and 43 and 345, but 123 sends it to 3. 3 is missed by 675, but 43 sends it to 4, that 4 is sent to 5 by the 345, and that is left alone by the 123 so 5 is spat out. 5 is sent to 6 by 675, and that 6 is left alone by the other ones so 6 is spat out. 6 is sent to 7 by 675, 7 is left alone by the rest so 7 is spat out 7 is sent to 5 by 675, 5 is left alone by 43, but sent to 3 by 345, that 3 is sent to 1 by 123, so 1 is spat out. stop. the first cylce is: (123567) that just leaves the 4. let us check 4 is left alone 675 does nothing. 43 sends it to 3, 345 sends it back to 4, and that is left alone by 123, so 4 is indeed sent to 4. so that element has order 6.

Yes, because I predict he's completely ignoring the definitions and using his own private meanings

You could show that n^r < r^n for a fixed r>1 and all n sufficiently large. Of course if you don't tell us what techniques you are allowed to adopt then we're going to struggle to find you any proof that is acceptable.

EH? You know its jsut the LCM, as in least common multiple, of the lengths of the disjoint cycles, so are you asking me to show you how to find the LCM or are you aksing me how to write a permutation as a product of disjoint cycles?

No it doesn't: Given any set S, let f be an injection from S to P(S), its power set. Define T to be { t in S : t not in f(t)} then T is not in the image of f - and this is more in the spirit of how Cantor wrote his proof originally (and is only one of few of the proofs he offered). That is all the diagonal argument is - there is nothing special about N and R in it, other than there is a nice explicit description that shows us how it works.

Integral test. In the second there are I'm sure many explanations of this, but it simply boils down to replacing the orginal with a Geometric Series. You ought to try using that hint to prove it yourself.

That isn;'t the lcm, that's the gcd. the only element in any group of order 1 is the identity as well.

To be honest I neither know nor care about constructible numbers. I was merely pointing out that in saying Cantor's argument doesn't apply to them you are making a misleading statement since all cantor does is show that there is no bijection between a set and its power set.

"Strictly speaking, infinite decimals ought not to be allowed at all since they are not constructible, so Cantor's proof would fall apart if we were forced to use only constructible numbers." In what sense are you being strict? Since any constructible number has an (infinitely long) decimal expansion, this is a very moot point. However, you are almost making the same mistake as Doron. Cantor's argument isn't about constructible numbers so your objections are off topic, and slightly irrelevant. We aren't forced to use constructible numbers, nor shoud we be.

Doron's arguments maybe sound in and of themself, though I seriously doubt it as no one has ever managed to make him explain anything of it in terms that are sound. I object to his presumptions of redefining (though without explicitly saying as what) terms that are well known, and doing something unmathematical with it based upon some of his personal preferences, to conclude that maths itself is flawed. Typical example: Cantor's diagonal argument is wrong since there is no such thing as an infinite "complete" set, where "complete" is not defined. The clincher in this is that in the alleged refutation of it he actually gives the original proof and doesn't manage to demonstrate where it is wrong. Everyone should remember, especially Daymare, that there is no such thing as absolute truth in mathematics. And if you don't believe that, D17, then can I suggest you examine Skolem's paradox - that there is a model of ZF in which all sets (including the power set of N) are countable. Countable though in the sense that although there is a bijection with N, that bijection may not be set in the model (and in the case of the power set, which isn't the power set you're thinking of, certainly isn't - Cantor's argument is still true). Of course I doubt you'll believe that functions are sets.

If you can give us one single mathematical object which is not equal to itself, in the correct context of equal, then perhaps we all should stop. However all you've offered is that an apple, or some other physical object, is not equal to itself, with some odd sense of equal. Consider the number sqrt(2), consider the smallest positive root in the reals to the polynomial x squared minus two. In what *mathematical* way are they not equal? Incidentally, the mathematical philsophy that prevails currently appears to be formalism, not idealism, or platonism. What ever the initial motivation to study was doesn't mean we shouildn't go beyond it. And you're still wrong about the motivation for 1=1.We are not saying physical objects are equal but the (well defined quantity when it exists) of two collections are equal. Ie if we started counting things one at a time in each collection starting at 1 we'd hit the same number when we came to the final object in each collection. That is all. We are not claiming anything like the objects are equal, so stop saying we are.

Pick generators of Sn, hint transpositions generate, all transpositions are conjugate.

how can p be in the centre? p is a number? Look up the class equation: the order of the group is the sum of the sizes of the conjugacy classes. All conjugacy classes have order a power of p, and there is at least one conjugacy class with a single element in it, so....

No, not as stands. Proving A4 has no subgroup of order six isn't going to help unless you have provem that the assumption G exists and isn't A4 implies that it must possess such. You don't appear to have done that.

Or, to put it in slightly different terms. If f is unformly continuous on some set S, it is unformly continuous on every subset of it. So x^2 is continuous on (0,1). R is closed, when thought of as a subset of R, and that certainly isn't bounded. The integers are a closed subset of R and aren't bounded.

Absolutely nothing of that kind is inplied: Any continuous function on a closed interval (or compact if you know what that is) is uniformly continuous.

Sort of but it's important that f is uniformly continuous. Uniform continuity means that when you do the formal e-d proof of continuity oat x, the d you choose can be independent of the point x. For instance the function 1/x does not have a continuation onto [0,1] even though it is continuous on (0,1). You see, the limit as x tends to 0 of 1/x doesn't exist, but it would if it were a uniformly continuous function.

so? what's that got to do with 1=1? this is a statement about a mathematical object, yours is about whatever philosophical problems you have with adopting a non-mathematical meaning for = for apples. The fact is that your interpretation for equals in the real world isn't anything to do with the mathematical one (equality is based upon, say the idea that S=T as sets iff x in S and x in T are equivalent propositions, or that [abstract] things are the same - have the same mathematical properties exactly).

and the point is that 1+1 is a statement about units in rings, or elements in the peano system. You're confusing reality and mathematics And how uninformed that you think that only those things that are real are mathematics. What is "real" anyway? Equality, addition, and so on are perfectly precise and the fact that there is only one unique instantation, physically and temporally of physical objects is neither here nor there. Evidently you do not know what an axiomatic system is, and a model of such, as is distinct from an application and model. Whatever the original motivation of numbers is (ie it counts the cardinality of sets [of apples]) has nothing to do with describing each apple uniquely. Did you not read, or just not understand, the simple explanation that when we say 1 apple plus one apple gives two apples we are talking about a property of collections of apples (how many there are, which is reasonably well understood, as apples are reasonably discrete objects) not about the apples themselves being in any sense "the same"? That is not what the equals sign means. "1" "exists" in the loose mathematical sense, independently of your use of it to say how many apples there are, since it is nothing to do with apples. I was trying to inform you of some of the things mathematics "is", since you evidently are ignorant of the subject. The idea that x=x is fundamental needs some explaining on your part. I'm happy with it - it's encoded in the ZF axioms after all, and I know what it means. You seemingly don't if you think 1 apple = 1 apple is both a mathematically sound statement (you're misinterpreting the "=" sign) and simultaneously about the apples being the same apple. You are confusing maths with its applications again. the only rule of thumb for a mathematical object is that it is not internally inconsistent, and is preferably non-vacuous. What exactly falls under the purview of mathematics is an endless topic of debate. If the maths we do has a use that models (and this is used in the sense you think it is, and not the sense above for axiomatics systems) things in the real world then so be it, especially if it is verifiable. The additive integers are a good way to count discrete well defined objects like apples. If I have two drops of water and I combine them how many drops do I have? When does it stop being a drop and become, say a puddle? That is not a problem of mathematics in and of itself but of your application of it to the real world. In short, you are misusing the equality symbol in a most depressing way. No one says that it means the two apples are the same. In fact that is one of the first things we beat out of students - misapplications of equals signs, until such time as they are sufficiently well read to start abusing them again.

It's the class equation again - the centre can never be trivial in a p-group.

Hint G is also normal.

2 things spring to mind: 1. ask the lecturer 2. when you talk of maximal subgroups (of a group) they are not unique, not even up to order, it just means H<G and whenever H<=K<=G then K =H or K=G. Saying it is the maximum subgroup (apart from being grammatically dubious if we're thinking of maximum as being an adjective) has the air of it being unique. but i must stress this is just an opinion - there are few conventions universally and uniquely adopted in maths. 3. but you should ignore 2 and go with 1. 4. no really you should go with 1.

It's not even as simple as epsilon-delta. Firstly convergence of functions on what space in what norm? Oh, and yes, I think it more than likely that this is going beyond what the poster is thinking of, but it is supposed to highlight the difficulty of answering that open ended question. Given any interval of the real line, say [0,1] the space of continuous real valued functions have an infinite number of norms on it: the L^n norms. none of which are complete. And there's L^{infinity}, which is complete (compact so continuous is uniformly continuous, and hence uniform limit of uniformly continuous functions is continuous. Not to mention the other innerproducts on it. Or do we mean R as the domain? Or are we talking about results independent of the norm, ie general metric space results. Or is it more along the lines of: if f is continuous and g is continuous show f+g is continuous (at a point)....