Definition List Modern Algebra, Fall 2011 Anders O.F. Hendrickson


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1 Definition List Modern Algebra, Fall 2011 Anders O.F. Hendrickson On almost every Friday of the semester, we will have a brief quiz to make sure you have memorized the definitions encountered in our studies. Below is a list of all the terms you might be asked to define, along with the sections of the book in which we will encounter them. Your answers do not need to be wordforword identical with the definitions presented here, but they must be exactly the same in meaning. After all, how can you prove theorems about something if you don t know exactly what it is? and In describing a function α, we use the symbol to connect the sets (the domain and codomain), as in α : S T. We use the symbol to indicate where a specific element goes; for example, g g 2 means α(g) = g 2. image The image of A S under α : S T is {α(a) : a A} and is written α(a). identity mapping The identity mapping of a set S is the function ι S : S S given by ι S (x) = x for all x S. equality of mappings Two mappings α and β are equal if they have the same domain and the same codomain and α(x) = β(x) for every x in the domain. onto The function α : S T is onto if α(s) = T ; in other words, for each t T there exists an s S such that α(s) = t. onetoone The function α : S T is onetoone if s 1 s 2 implies α(s 1 ) α(s 2 ) 2 2 for all s 1, s 2 S. Equivalently, if s 1, s 2 S and α(s 1 ) = α(s 2 ), we can conclude that s 1 = s 2. composition Let α : S T and β : T U. The composition of α and β is the function β α : S U s β(α(s)). inverse function Let α : S T. We say β : T S is an inverse of α if β α = ι S and α β = ι T. Cartesian product The Cartesian product of sets S and T is the set of ordered pairs {(s, t) : s S, t T }, which is written S T.
2 Definition List Modern Algebra, Fall 2011 Page 2 of closed We say that a set S is closed under if a b S for all a, b S. operation A (binary) operation on the set S is a function from S S S. In other words, it is a rule for combining two elements of S to get a third one. associative The operation on set S is associative if a (b c) = (a b) c for all a, b, c S. identity element An identity element for a set S and an operation is an element e S such that e x = x and x e = x for all x S. inverse element Let be an operation on set S with identity e. We say b S is an inverse of a S if a b = e and b a = e. commutativity An operation on set S is called commutative if a b = b a for all a, b S. M(S) Let S be a set. Then M(S) is the set of all mappings from S to S. group A group is a set G with an operation that satisfies these four axioms: 1. a b G for all a, b G. (Closure) 2. There exists an identity element e G. (Identity) 3. For every a G there exists some b G such that a b = b a = e. (Inverses) 4. a (b c) = (a b) c for all a, b, c G. (Associativity) abelian A group G with operation is abelian if a b = b a for all a, b G. In other words, the group s operation is commutative. order of a group The order of a group is the number of elements it has. It is written G. permutation A permutation of a set S is a map from S S that is onetoone and onto. symmetric group Given a set S, the symmetric group Sym(S) is the set of all permutations of S; the group operation is composition. permutation group A permutation group is a group whose elements are permutations of some set, and whose operation is composition. cycle A permutation of the form (a 1 a 2 a 3... a k ) is called a kcycle or simply a cycle.
3 Definition List Modern Algebra, Fall 2011 Page 3 of conjugate If G is a group and a, b G, then the conjugate of a by b is bab 1. subgroup A subgroup of the group G is a subset of G that is itself a group under the same operation as G. transposition A transposition is a 2cycle, that is, a permutation that just swaps two elements. even permutation A permutation is called even if it can be written as the product of an even number of transpositions. odd permutation A permutation is called odd if it can be written as the product of an odd number of transpositions. alternating group The alternating group A n is the set of all even permutations in S n. reflexive A relation on set S is called reflexive if a a for all a S. symmetric A relation on set S is called symmetric if a b implies b a for all a, b S. transitive Let be a relation on set S. Suppose that for all a, b, c S, if a b and b c, then a c; in this case, we say is transitive. equivalence relation An equivalence relation is a relation that is symmetric, reflexive, and transitive. equivalence class Suppose we have an equivalence relation on a set S, and let a S. equivalence class of a is [a] = {x S : x a}. Then the partition A partition P of a set S is a collection of nonempty subsets of S that are pairwise disjoint such that S is the union of all the sets in P. class representatives If we have an equivalence relation on a set S, a set of equivalence class representatives is a set that contains exactly one element from each equivalence class. divisibility We say an integer m is divisible by n if and only if m = nk for some integer k. In this case, we write n m. congruent We say two integers a and b are congruent modulo n if and only if n (a b). This is written as a b (mod n).
4 Definition List Modern Algebra, Fall 2011 Page 4 of Z n Let n be a positive integer. Then Z n is the set of equivalence classes of the integers under the relation congruence mod n. The operation is [a] [b] = [a + b]. greatest common divisor If 0 a, b Z, then their greatest common divisor is the unique positive integer d such that 1. d divides both a and b (i.e., it is a common divisor), and 2. every common divisor of a and b divides d. 2 2 This number is denoted (a, b). coprime Integers a and b are called coprime or relatively prime if their greatest common divisor is 1. least common multiple If 0 a, b Z, then their least common multiple is the unique positive integer m such that 1. both a and b divide m (i.e., it is a common multiple), and 2. m divides every common multiple of a and b. 3 This number is denoted [a, b]. standard form Let n > 1 be an integer. The standard form of n is the unique way of writing 3 U n e n = p 1 e2 e 1 p 2 p k k such that p 1 < p 2 < < p k are prime numbers and e 1,..., e k are positive integers. Let n be a positive integer. Then U n is a group defined as follows: working modulo n, U n = {[k] : 1 k < n and (k, n) = 1} The group operation is multiplication mod n, denoted. Euler ϕfunction Let n > 1 be an integer. Then ϕ(n) is defined to be the number of positive integers that are less than n and relatively prime to n. This is called the Euler ϕfunction. cyclic group A group G is called cyclic if G = a for some a G. order of an element Let a be an element of some group. The order of n is the smallest positive integer n such that a n = e. It is written o(a).
5 Definition List Modern Algebra, Fall 2011 Page 5 of S Let S be a subset of a group G. Then the subgroup generated by S, written S, is the unique smallest subgroup of G that contains S. In other words, if H is a subgroup containing S, then H must also contain S. direct product Let A and B be groups. Then their direct product A B is the set of ordered pairs {(a, b) : a A, b B} under the operation (a 1, b 1 )(a 2, b 2 ) = (a 1 a 2, b 1 b 2 ). coset Let H be a subgroup of G and let a G. Then the right coset of H containing a is Ha = {ha : h H}. 7 8 (Left cosets are defined similarly.) index Let H be a subgroup of G. Then the index of H in G, written [G : H], is the number of right cosets of H in G. homomorphism Let G be a group with operation and H be a group with operation #. function θ : G H is a homomorphism if Then a θ(a b) = θ(a) # θ(b) for all a, b G. isomorphism A function θ : G H is an isomorphism if it is onetoone, onto, and a homomorphism. isomorphic We say two groups G and H are isomorphic if there exists an isomorphism θ : G H. In this case, we write G H or G = H. kernel Let θ : G H be a homomorphism. The kernel of θ is ker θ = {g G : θ(g) = e H }. normal Let N be a subgroup of G. Then N is normal if In this case, we write N G. gng 1 N for all n N and g G.
6 Definition List Modern Algebra, Fall 2011 Page 6 of quotient group Let N G. The quotient group or factor group G/N is the set of all right cosets of N in G. The operation is (Na)(Nb) = N(ab). natural homomorphism Let N G. Then the natural homomorphism from G to G/N is η : G G/N a Na. simple group A simple group is a nontrivial group with only two normal subgroups, namely itself and {e}. group action Let G be a group and S be a set. We say G acts on S if for every s S and g G, an element s g S is defined such that for all s S and g, h G, (s g) h = s (gh) s e = s faithfully We say G acts faithfully on S when, if for some g G we have s g = s for all s S, then g = e; i.e., the only element of G that acts like the identity is e itself. orbit Let G act on S. Let s S. The orbit of s under G s action is O s = {s g : g G}. Fix(g) Let G act on S, and let g G. Then the set Fix(g) = {s S : s g = s}. stabilizer Let G act on S, and let s S. The stabilizer of s is the subgroup G s = {g G : s g = s}. Sylow psubgroup Let G be a finite group and let p be prime. Let p a be the largest ppower that divides G. Then any subgroup of G of order p a is called a Sylow psubgroup of G. Syl p (G) Let G be a finite group and p be a prime. Then Syl p (G) is the set of all Sylow p subgroups of G. n p (G) Let G be a finite group and let p be prime. Then n p (G) is the number of Sylow psubgroups of G, and is sometimes abbreviated as just n p. ring A ring is a set R with two operations, + and, such that
7 Definition List Modern Algebra, Fall 2011 Page 7 of 7 1. R is an abelian group under addition. 2. Multiplication is associative. 3. For all a, b, c R, a(b + c) = ab + ac. 4. For all a, b, c R, (a + b)c = ac + bc The additive identity is usually written 0. commutative ring A commutative ring is a ring in which multiplication is commutative. unity A unity of a ring R is multiplicative identity; that is, it is an element e R such that ex = xe = x for all x R. We often write the unity as 1. zero divisor If x and y are nonzero elements of a commutative ring R such that xy = 0, we say that x and y are zero divisors. integral domain An integral domain is a commutative ring with unity with no zero divisors. subring We say that S is a subring of a ring R if S is a subset of R and S is a ring with the same operations as R. field A field is a commutative ring with unity such that each nonzero element has a multiplicative inverse. Equivalently, a field F is a ring with that F {0} is an abelian group under multiplication.
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