PUMaC 2015 · 加试 · 第 3 题

Problem 3.6 (Odd Order; 5 ) . Let G be a finite group. Prove that an element g has odd order in G if and k 2 only if for every positive integer k , there exists some element h ∈ G such that ( h ) = g . (Note that we k k do not specify G as abelian; it is sufficient to take G a finite group.) Proof. (Note to graders: many proofs of this may get very long and overly convoluted. Make sure that they do not make any more assumptions beyond the given that G is finite). k ( ) 2 k 2 l +1 2( l +1) ( l +1) ( ⇒ ) Suppose g = 1 is the order. Then g = g . By induction, we see that for all k ∈ N , g = g . l ( ⇐ ) Note that an element g has odd order iff some odd power of g is 1. So suppose | G | = 2 · x where x is l 2 x odd. Then look at g = h . We see g = 1, and g has odd order. l Hopefully you have found these problems illuminating (hint: you may see these things again). However, to end this section, we would just like to present some standard material that we think is enriching to experience. Suppose we have two groups G and H , and we create a function f between them f : G → H . But we don’t want any old functions; we want functions that somehow show that the structure of G and H have similarities. For example, examine Z and 2 Z as additive groups (2 Z is notation for the set of all even numbers). Any addition done with the even numbers is linked to addition done with regular numbers. For example, 2 + 46 = 48 ⇔ 2(1) + 2(23) = 2(24) . We want functions between groups to somehow reflect the similarities between them. Thus we impose more conditions on f . Definition 10. Let G and H be groups with a function f : G → H . This function f is called a homomor- phism if: • For all a, b ∈ G , f ( a ∗ b ) = f ( a ) ∗ f ( b ) ( ∗ and ∗ represent the operations of G and H respectively). G H G H For example, let G = Z , H = 2 Z , and let f ( a ) = 2 a . We encourage you to check that this is indeed a homomorphism, and that it has the property we wanted: that it represents the relationship between the structure of G and H . Definition 11. Let f : G → H be a homomorphism. If every element of H is the image of some element of G , then f is surjective and is a surjection. Definition 12. Let f : G → H be a homomorphism. If the only element of G that is mapped to e , the H identity element of H , is e , then f is injective and is an injection. G Definition 13. If a group homomorphism is both injective and surjective (if it satisfies both, then it is also called bijective), then it is called an isomorphism. You may want to convince yourself that in the example above, the homomorphism is indeed bijective. In general, the same terminology applies to general functions. If a function f : A → B has the property that ′ ′ ′ ∀ b ∈ B , ∃ a ∈ A such that f ( a ) = b (surjective) and that ∀ a, a ∈ A , f ( a ) = f ( a ) = ⇒ a = a (injective), 9 PUMaC 2015 Power Round Section 3 page 10 then f is bijective (this is an adjective) and is a bijection (this is a noun). Next, we introduce the idea of mashing groups together. Recall that a number line can be represented as 2 R . A coordinate plane is similarly represented by the notation R or R × R . This is called a direct product, and 2 elements of R are coordinate pairs ( a, b ) where a and b both come from R . Well, this is something we can do with groups as well. For example, taking two groups G and H , we denote G × H := { ( g, h ) : g ∈ G, h ∈ H } . The group operation on this new set is the combination of the operations of G and H component-wise respectively. One fact that we ask the reader to convince themself (this is clearly grammatically not “correct,” but the author strongly believes this should be an official singular, gender-neutral pronoun) is that G × H is itself a group. While this isn’t an official problem, you should convince yourself that this works because it’s necessary to understand for the next part. We first introduce a definition. We will explore this topic more thoroughly in section 7, but the definition is sufficient here for now. Definition 14. Let R be a set of elements with a closed binary operation we call addition such that { R, + } is an additive, commutative group. Furthermore, suppose R admits another closed binary operation · that we can call multiplication; it is commutative, and ∀ a, b, c ∈ R , a · ( b · c ) = ( a · b ) · c ; a · ( b + c ) = a · b + a · c ; and ( b + c ) · a = b · a + c · a . Finally, there exists a multiplicative identity we call 1 such that for all a ∈ R , a · 1 = 1 · a = a . Then R is called a ring. Definition 15. Let R be a ring. Then GL ( R ) is called the general linear group, and denotes the group of n n × n invertible matrices with elements in R as a group under multiplication. There is yet another way to make a group from two smaller groups. As a generalization of the direct product, we introduce the semidirect product. We start with a group G and a group H , where H is a group of functions that act on elements of G . For example, let G be the set of vectors Z × Z , and let H be the set of matrices GL ( Z ). 2 In other words, G is a set of vectors of the form ( a, b ) where a and b are both integers, and H is the set of invertible 2 × 2 matrices with integer coefficients (invertible under multiplication). Elements of H are indeed functions that act on elements of G ; for example if h ∈ H and g ∈ G , then h · g is another vector. Thus we can define a semidirect product as follows. Proposition 16. Let the semidirect product setwise be defined as G o H := { ( g, h ) : g ∈ G, h ∈ H } where H is a group of functions that acts on G (yes, groups may have functions as elements). Let ∗ and ∗ denote the g h group operations of G and H respectively. Then the group operation on this set for ( g , h ) , ( g , h ) ∈ G o H 1 1 2 2 is defined by ( g , h ) ∗ ( g , h ) := ( g ∗ h ( g ) , h ∗ h ) . 1 1 2 2 1 g 1 2 1 h 2 G o H is a group. Thus in our example here, we can define a group ( Z × Z ) o GL ( Z ) (this is an example of an affine general 2 linear group as we shall see later). Here is an example operation: ([ ] [ ]) ([ ] [ ]) ([ ] [ ] [ ] [ ] [ ]) 2 − 5 2 1 1 0 2 − 5 2 1 − 5 2 1 0 , ∗ , = + · , · 0 2 − 1 5 0 1 0 2 − 1 5 2 − 1 0 1 ([ ] [ ] [ ] [ ]) 2 5 − 5 2 1 0 = + , · 0 − 3 2 − 1 0 1 ([ ] [ ]) 7 − 5 2 = , . − 3 2 − 1 10 PUMaC 2015 Power Round Section 4 page 11 2 Later on when we revisit this affine general linear group, we use this notation AGL ( R ) to denote ( R ) o 2 GL ( R ) where R is a general ring. We will go in depth about this later. With this, we turn now to the topic 2 of elliptic curves. 4 Elliptic Curves Elliptic curves are integral (hah! it’s a pun!) to mathematics, and in fact have even higher generalizations called varieties. For the purposes of this power round, we define an elliptic curve as follows. Definition 17. An elliptic curve E is the curve satisfying an equation of the form 2 3 2 E : y + a xy + a y = x + a x + a x + a , 1 3 2 4 6 where the coefficients a are INTEGERS (can’t stress this enough :p). i The strange coefficient numberings are of historical significance. The reason that elliptic curves are interesting is that there is a natural group on the set of points on them. Very important remarks : for the entirety of this power round, we take coefficients of elliptic curves to be integers. However, we may allow points on the curve to be rational or something else. Unless oth- erwise specified, any elliptic curve has integer coefficients and any points we consider on it are rational points. Now, if you’re the average person looking at this equation, you may be slightly disgusted by how unwieldy it looks; come on, that xy term looks atrocious. So let’s get rid of it.