GRE Math Subject Test key terms

A group whose binary operation is commutative, so a·b = b·a for all elements. Every cyclic group is Abelian, but the symmetric group on three or more letters is not.

A series converges absolutely when the series of absolute values of its terms converges. Absolute convergence implies ordinary convergence, but the converse fails, as with the alternating harmonic series.

A linearly independent spanning set for a vector space. Every vector has a unique expression as a finite linear combination of basis elements, and all bases of a finite-dimensional space have the same size.

A sequence whose terms eventually become arbitrarily close to each other: for every positive epsilon there is an index beyond which all pairwise distances are below epsilon. In the real numbers every such sequence converges.

For a square matrix or linear operator, the polynomial det(A minus lambda times the identity). Its roots are the eigenvalues, counted with algebraic multiplicity, and its degree equals the dimension of the space.

A topological property: every open cover admits a finite subcover. In Euclidean space a set is compact exactly when it is closed and bounded, which is the form usually tested.

A series converges but the series of absolute values diverges. Such series can be rearranged to converge to any chosen real value or to diverge, by the Riemann rearrangement theorem.

A function is continuous at a point when the limit there equals the function value. Equivalently, preimages of open sets are open, and on a compact set a continuous function attains maximum and minimum values.

A group generated by a single element, so every element is a power of that generator. Cyclic groups are Abelian, and every subgroup of a cyclic group is cyclic.

A linear operator is diagonalizable when some basis consists entirely of eigenvectors. This holds exactly when the minimal polynomial has distinct linear factors, and it fails for defective matrices.

A function is differentiable at a point when its difference quotient has a finite limit there. Differentiability implies continuity, but continuity does not imply differentiability, as the absolute value function at zero shows.

The number of vectors in any basis of a finite-dimensional vector space. It equals the number of rows of the row-reduced echelon form that contain pivots.

A scalar lambda for which some nonzero vector v satisfies Av = lambda v. Eigenvalues are roots of the characteristic polynomial and can be complex even for real matrices.

A repeated division procedure that computes the greatest common divisor of two integers. Each remainder is expressed in terms of the previous pair, and the process terminates at the gcd.

For a prime p and integer a not divisible by p, a raised to p minus one is congruent to 1 modulo p. Equivalently, a raised to p is congruent to a for every integer a.

A commutative ring with identity in which every nonzero element has a multiplicative inverse, so addition, subtraction, multiplication, and division behave as in the rationals. The reals and integers modulo a prime are fields.

The two-part theorem linking differentiation and integration: the integral of a derivative recovers the function up to a constant, and the derivative of an accumulation integral recovers the integrand.

A set with an associative binary operation, an identity element, and inverses for every element. Groups need not be commutative, and the symmetric group is the standard noncommutative example.

In Euclidean space, a subset is compact exactly when it is closed and bounded. This characterization is used constantly to justify extrema and uniform continuity on intervals.

For a finite group, the order of any subgroup divides the order of the group. A consequence is that the order of any element divides the group order, so a to the power of the group order is the identity.

A set of vectors is linearly independent when no nontrivial linear combination of them equals zero. Equivalently, no vector in the set lies in the span of the others.

If a function is continuous on a closed interval and differentiable inside it, then some interior point has derivative equal to the average slope across the interval. It underlies many inequality and monotonicity arguments.

For a linear operator, the monic polynomial of least degree that annihilates it. It divides the characteristic polynomial, and its distinct linear factors determine diagonalizability.

Arithmetic on remainders after division by a fixed positive integer, where two integers are equivalent when their difference is a multiple of that modulus. Inverses exist exactly for residues coprime to the modulus.

A subgroup invariant under conjugation by every element of the containing group. Normal subgroups are exactly the kernels of homomorphisms, and they allow construction of quotient groups.

A nonconstant analytic function maps open sets to open sets. The maximum modulus principle follows from it, since the modulus of a nonconstant analytic function cannot attain an interior maximum.

A complex function that is differentiable in a neighborhood of each point of its domain. Such a function has a convergent power series locally and satisfies the Cauchy-Riemann equations.

An integer greater than one whose only positive divisors are one and itself. Euclid proved infinitely many exist, and the prime number theorem describes their density near a bound.

For a linear map from an n-dimensional space, the rank plus the nullity equals n. Rank is the dimension of the image and nullity the dimension of the kernel.

A convergence test comparing the limit of consecutive term ratios to one. The series converges absolutely when the limit is below one and diverges when it is above one.

The integral of a meromorphic function around a closed contour equals two pi i times the sum of its residues at enclosed poles. It converts many real integrals into residue computations.

The limit of Riemann sums over partitions as mesh size shrinks. A bounded function on a closed interval is Riemann integrable exactly when its discontinuities form a set of measure zero.

If a function is continuous on a closed interval, differentiable inside it, and equal at the endpoints, then its derivative vanishes somewhere inside. It is the special case underlying the mean value theorem.

The set of all finite linear combinations of a collection of vectors. It is always a subspace, and it equals the smallest subspace containing the collection.

The power series formed from the derivatives of a smooth function at a point. It equals the function on an interval when the remainder tends to zero, as for exponential and sine.

A sequence of functions converges uniformly when the supremum distance between term and limit tends to zero. Uniform limits preserve continuity, whereas pointwise limits need not.

The count of integers from one to n that are coprime to n. It is multiplicative, and it gives the size of the group of units modulo n.

The largest positive integer dividing both of two given integers. It can be written as an integer combination of them, a fact used to solve linear congruences.