Math 250A: Groups, rings, and fields. H. W. Lenstra jr. 1. Prerequisites This section consists of an enumeration of terms from elementary set theory and algebra. You are supposed to be familiar with their definitions and basic properties. Set theory. Sets, subsets, the empty set , operations on sets (union, intersection, ; product), maps, composition of maps, injective maps, surjective maps, bijective maps, the identity map 1X of a set X, inverses of maps. Relations, equivalence relations, equivalence classes, partial and total orderings, the cardinality #X of a set X. The principle of math- ematical induction. Zorn's lemma will be assumed in a number of exercises. Later in the course the terminology and a few basic results from point set topology may come in useful. Group theory. Groups, multiplicative and additive notation, the unit element 1 (or the zero element 0), abelian groups, cyclic groups, the order of a group or of an element, Fermat's little theorem, products of groups, subgroups, generators for subgroups, left cosets aH, right cosets, the coset spaces G=H and H G, the index (G : H), the theorem of n Lagrange, group homomorphisms, isomorphisms, automorphisms, normal subgroups, the factor group G=N and the canonical map G G=N, homomorphism theorems, the Jordan- ! H¨older theorem (see Exercise 1.4), the commutator subgroup [G; G], the center Z(G) (see Exercise 1.12), the group Aut G of automorphisms of G, inner automorphisms. Examples of groups: the group Sym X of permutations of a set X, the symmetric group S = Sym 1; 2; : : : ; n , cycles of permutations, even and odd permutations, the alternating n f g group A , the dihedral group D = (1 2 : : : n); (1 n 1)(2 n 2) : : : , the Klein four group n n h − − i V , the quaternion group Q = 1; i; j; ij (with ii = jj = 1, ji = ij) of order 4 8 { g − − 8, additive groups of rings, the group Gl(n; R) of invertible n n-matrices over a ring R. × Occasionally the structure theorem of finite abelian groups and finitely generated abelian groups will be assumed known. Ring theory. Rings, subrings, homomorphisms. We adopt the convention that the existence of a unit element 1 is part of the definition of a ring. Likewise, subrings of a ring R are required to contain the unit element of R, and ring homomorphisms are required to map 1 to 1. Products of rings, zero-divisors, units, the group R∗ of units of a ring R. Ideals, operations on ideals (sum, intersection, product), generators for ideals, principal ideals, the factor ring R=a, congruence modulo an ideal. Domains, prime ideals, maximal ideals. Principal ideal rings, unique factorization domains. Examples: the ring Z of integers, the ring Z=nZ of integers modulo n, the zero ring, polynomial rings R[X1; : : : ; Xn], the endomorphism ring End A of an abelian group A (see Exercise 1.23), the ring M(n; R) of n n-matrices over a ring R, the division ring (or skew × field) H of quaternions, the group ring R[G] of a group G over a ring R (see Exercise 1.25), 1 the ring R[[X]] of formal power series over a ring R. (see Exercise 1.26). Field theory. Fields, the field Q(R) of fractions of a domain R, the prime field, the characteristic char k of a field k, algebraic and transcendental elements over a field, alge- α braic and finite extensions, the degree [l : k], the irreducible (or minimum) polynomial fk α of an element α algebraic over k, the isomorphism k(α) ∼= k[X]=fk k[X], splitting fields. Examples: the fields Q, R, and C of rational, real, and complex numbers, respec- tively, the field Fp = Z=pZ (for p prime), finite fields Fq (for q a prime power), the field k(X1; : : : ; Xn) of rational functions in n indeterminates over a field k. 1. Exercises 1.1 (Left axioms). Let G be a set, 1 G an element, i: G G a map, and m: G G G 2 ! × ! another map; instead of m(x; y) we write xy, for x, y G. 2 (a) Suppose that for all x, y, z G one has 2 1x = x; i(x)x = 1; (xy)z = x(yz): Prove that G is a group with multiplication m. (b) Suppose that for all x, y, z G one has 2 x1 = x; i(x)x = 1; (xy)z = x(yz): Does it follow that G is a group with multiplication m? Give a proof or a counterexample. 1.2. Let u, U, v, V be subgroups of a group G. Assume that u is a normal subgroup of U and that v is a normal subgroup of V . Prove that u(U v) is a normal subgroup of \ u(U V ), that (u V )(U v) is a normal subgroup of U V , that (u V )v is a normal \ \ \ \ \ subgroup of (U V )v, and that there are group isomorphisms \ u(U V )= u(U v) = (U V )= (u V )(U v) = (U V )v= (u V )v : \ \ ∼ \ \ \ ∼ \ \ r 1.3. A normal tower of a group G is a sequence (ui)i=0 of subgroups ui of G for which r is a non-negative integer, each ui 1 is a normal subgroup of ui (for 0 < i r), and u0 = 1 , − r t ≤ f g ur = G. A refinement of a normal tower (ui)i=0 is a normal tower (wj)j=0 with the property that there is an increasing map f: 0; 1; 2; : : :; r 0; 1; 2; : : :; t such that for all i one f r g ! sf g has ui = wf(i). Two normal towers (ui)i=0 and (vj)j=0 are called equivalent if there is a bijection g: 1; 2; : : :; r 1; 2; : : : ; s such that for all i one has ui=ui 1 = vg(i)=vg(i) 1. f g ! f g − ∼ − Prove the Schreier refinement theorem: any two normal towers of a group have equivalent refinements. 1.4. A group is called simple if it has exactly two normal subgroups (namely 1 and the r f g group itself). A composition series of a group is a normal tower (ui)i=0 for which every group ui=ui 1 is simple. − 2 (a) Prove the Jordan-H¨older theorem: any two composition series of a group are equivalent. (b) Exhibit a group that does not have a composition series. (c) Classify all simple abelian groups. 1.5. (a) Prove that, up to isomorphism, there are precisely 9 groups of order smaller than 8. Which are they? Formulate the theorems that you use. (b) How many pairwise non-isomorphic groups of order 8 are there? Prove the cor- rectness of your answer. 1.6. Let H, N be groups, and let : H Aut N be a group homomorphism. Instead of ! (h)(x) we shall write hx, for h H, x N. 2 2 (a) Prove that the set N H with the operation × h (x; h) (y; h0) = (x y; hh0) · · is a group. This group is called the semidirect product of H and N (with respect to ), notation: N o H. (b) Prove that N may be viewed as a normal subgroup of N oH, and that (N oH)=N is isomorphic to H. (c) Is the direct product a special case of the semidirect product? 1.7. (a) Let G be a group, let H be a subgroup of G, and let N be a normal subgroup of G. Suppose that the composition of the inclusion H G with the natural map G G=N is ⊂ ! an isomorphism H ∼ G=N. Prove that there is a group homomorphism : H Aut N −! ! such that the map N o H G sending (x; h) to xh is a group isomorphism. ! (b) Let N and N be two normal subgroups of a group G with N N = 1 . Prove 1 2 1 \ 2 f g that for all x N , y N one has xy = yx. 2 1 2 2 1.8. Prove that S is isomorphic to V o S with respect to some isomorphism : S 4 4 3 3 ! Aut V4. 1.9. A subgroup H of a group G is called characteristic (in G) if for all automorphisms ' of G one has 'H = H. (a) Prove that any characteristic subgroup of a group is normal. (b) Give an example of a group and a normal subgroup that is not characteristic. 1.10. (a) Let H be a subgroup of a group G, and let J be a characteristic subgroup of H. Prove: if H is normal in G then J is normal in G, and if H is characteristic in G then J is characteristic in G. (b) If H is a normal subgroup of a group G, and J is a normal subgroup of H, does it follow that J is normal in G? Give a proof or a counterexample. 3 1.11. (a) Prove that there is a non-abelian group G of order 8 with the property that every subgroup of G is normal. (b) Does there exist a group that has exactly one subgroup that is not normal? Give an example, or prove that no such group exists. 1.12. Let G be a group. By Z(G) we denote the center of G, i. e.: Z(G) = x G : for all y G one has xy = yx : f 2 2 g (a) Prove that the commutator subgroup [G; G] and the center Z(G) of G are char- acteristic subgroups of G.