ALGEBRA PRELIM - DEFINITIONS AND THEOREMS HARINI CHANDRAMOULI Group Theory Group A group is a set, G, together with an operation · that must satisfy the following group axioms: • Closure. 8 a; b 2 G, a · b 2 G. • Associativity. 8 a; b; c 2 G,(a · b) · c = a · (b · c). • Identity Element. 9 e 2 G such that 8a 2 G, e · a = a · e = a. Such an element is unique. • Inverse Element. 8 a 2 G, 9 a−1 2 G such that a · a−1 = a−1 · a = e. p-Sylow Subgroup Let pe be the largest power of p dividing the order of G.A p-Sylow subgroup (if it exists) is a subgroup of G of order pe. Sylow Theorems Theorem 1. For any prime factor p with multiplicity n of the order of a finite group G, there exists a p-Sylow subgroup of G, of order pn. Theorem 2. Give a finite group G and a prime number p, all p-Sylow subgroups of G are conjugate to each other, i.e. if H and K are p-Sylow subgroups of G, then there exists an element g 2 G with g−1Hg = K. Theorem 3. Let p be a prime factor with multiplicity n of the order of a finite group G, so that the order of G can be written as pnm where n > 0 and p does not divide m. Let np be the number of p-Sylow subgroups of G. Then the following hold: (1) np divides m, which is the index of the p-Sylow subgroup in G (2) np ≡ 1 (mod p) (3) np = jG : NG(P )j where P is any p-Sylow subgroup of G and NG denotes the normalizer Lagrange's Theorem For any finite group G, the order of every subgroup H of G divides the order of G. Note: Every group of prime order is cyclic. Theorem Every group of even order contains an element of order 2. 1 2 HARINI CHANDRAMOULI Fundamental Theorem of Finite Abelian Groups Every finite abelian group is an internal group direct product of cyclic groups whose orders are prime powers. Theorem Let G be a group and p be a prime number. If jGj = p2, then G is abelian. Theorem Let H and K be subgroups of G. Then, jHj · jKj jHKj = : jH \ Kj Index of a Subgroup The index of a subgroup H in a group G is the \relative size" of H in G; equivalently, the number of \copies" (cosets) of H that fill up G. ! If N is a normal subgroup of G, then the index of N in G is also equal to the order of the quotient group G=N, since this is defined in terms of a group structure on the set of cosets of N in G. Action of G on X Let X be a set and G a group. An action of G on X is a map ∗ : G × X ! X such that (1) ex = x; 8 x 2 X. (2) (g1g2)(x) = g1(g2x); 8 x 2 X and 8 g1; g2 2 G. Theorem Let p be the smallest prime dividing the order of a group G. If H is a subgroup of G with index p, then H is normal. Theorem Let G be a group and H, K be subgroups of G. Note that the product set HK is defined as HK := fhk j h 2 H; k 2 Kg : If H is normal in G, then HK is a subgroup of G. Moreover, if both H and K are normal in G, then HK is a normal subgroup of G. Quotient Group Let H be a normal subgroup of G. Then the cosets of H form a group G=H under the binary operation (aH)(bH) = (ab)H. The group G=H is the quotient group or factor group of G by H. th Dihedral Group The n dihedral group, D2n, is the group of symmetries of the regular n-gon. ! A regular polygon with n sides has 2n different symmetries: n rotational symmetries, and n reflection symmetries. The associated rotations and reflections makeup the dihedral group D2n. ! The composition of two symmetries of a regular polygon is again a symmetry of this object, giving us the algebraic structure of a finite group. ! The dihedral group with two elements, D2, and the dihedral group with four elements, D4, are abelian. For all other n, D2n is not abelian. ALGEBRA PRELIM - DEFINITIONS AND THEOREMS 3 Non-Abelian Group of order n3: Let n 2 N; n ≥ 2. A classical example of a non-abelian group of order n3 is the Heisenberg Group: 80 1 9 1 a c < = H(n) = @0 1 bA a; b; c 2 Z=nZ : : 0 0 1 ; We see that the identity matrix is in H(n) and the inverses are given by −1 01 a c1 01 −a −ab − c1 @0 1 bA = @0 1 −b A : 0 0 1 0 0 1 Theorem GLn(F ) is the general linear group of degree n over a field F with n×n invertible matricies, together with the operation of ordinary matrix multiplication. Let F be a finite field with q elements. Then the order of GLn(F ) is given as follows: n n n 2 n n−1 jGLn(F )j = (q − 1)(q − q)(q − q ) ··· (q − q ): Semidirect Product Let H and K be subgroups of a group G such that H C G. Suppose that α : K ! Aut(H) is a homomorphism between the group K and the automorphism group of the group H. Then the semidirect product of H and K determined by α, denoted by H oα K, is the set H × K equipped with the binary operation 0 0 0 0 (h; k)(h ; k ) = (hαk(h ); kk ): Theorem The semidirect product H oα K is a group under binary operation given above. Theorem Suppose that H and K are subgroups of G satisfying (1) H \ K = feg. (2) H C G. (3) HK = G. −1 Let α : K ! Aut(H) be given by αk(h) = khk be the automorphism of H given by the restriction of the conjugation operator to K, acting on H. Then the map H oα K ! G given by (h; k) 7! hk is an isomorphism. Theorem Let p be a prime number. Then, ∼ × Aut (Z=pZ) = (Z=pZ) : First Isomorphism Theorem Let G and H be groups, let ' : G ! H be a homomorphism. Then: (1) The kernel of ' is a normal subgroup of G. (2) The image of ' is a subgroup of H, and (3) The image of ' is isomorphic to the quotient group G= ker('). In particular, if ' is surjective, then H is isomorphic to G= ker('). 4 HARINI CHANDRAMOULI Rings and Fields Ring A ring is a set, R, equipped with binary operations + and · satisfying the following ring axioms: • R is an abelian group under addition. That is, (R; +) satisfies the group axioms and in addition, a + b = b + a holds 8 a; b 2 R. • R has a multiplicative identity and multiplication is associative. That is, (a · b) · c = a · (b · c) holds 8 a; b; c 2 R and 8 a 2 R, 9 1 2 R such that 1 · a = a · 1 = a. • Multiplication is distributive with respect to addition. That is, 8a; b; c 2 R, a·(b+c) = (a · b) + (a · c) holds (left distributivity) and (b + c) · a = (b · a) + (c · a) holds (right distributivity). Field A field is a set, F , equipped with two operations, + and ·, satisfying the following axioms: • Closure of F under addition and multiplication. That is 8 a; b 2 F , a + b 2 F and a · b 2 F . • Associativity of addition and multiplication. That is, 8 a; b; c 2 F , a + (b + c) = (a + b) + c and a · (b · c) = (a · b) · c. • Commutativity of addition and multiplication. That is, 8 a; b 2 F , a + b = b + a and a · b = b · a. • Existence of additive and multiplicative identity elements. That is, 8 a 2 F , 9 0 2 F and 9 1 2 F such that 0 + a = a and 1 · a = a. • Existence of additive and multiplicative inverses. That is, 8 a 2 F , 9 − a 2 F and 9 a−1 2 F such that a + (−a) = 0 and a · a−1 = 1. • Distributivity of multiplication over addition. That is, 8 a; b; c 2 F , a · (b + c) = (a · b) + (a · c). Integral Domain An integral domain is a commutative ring with multiplicative identity and no divisors of 0. Unique Factorization Domain A unique factorization domain (UFD) is an integral do- main in which every nonzero, noninvertible element has a unique factorization, that is, a decomposition as the product of prime elements or irreducible elements. Euclidean Domain Let R be an integral domain. A Euclidean function on R is a function f from Rn f0g to the non-negative integers such that if a; b 2 R and b 6= 0, then there are q; r 2 R such that a = bq + r and either r = 0 or f(r) < f(b). A Euclidean domain is an integral domain which can be endowed with at least one Euclidean function. ! Note that a particular Euclidean function f is NOT part of the structure of a Euclidean domain. In general, a Euclidean domain will admit many different Euclidean functions. ! Examples: Z - the ring of integers, K[x] - the ring of polynomials over a field K ALGEBRA PRELIM - DEFINITIONS AND THEOREMS 5 ! Every ideal in a Euclidean domain is principal.