IIT JAM MA key terms

A set together with a binary operation that is closed, associative, has an identity element, and gives every element an inverse. Groups model symmetry and are the central object of abstract algebra.

A nonempty subset of a group that is itself a group under the same operation. It must contain the identity, be closed under the operation, and contain inverses of its elements.

For a finite group, the order of any subgroup divides the order of the group. Consequently the order of any element divides the group order, a standard counting tool in finite group problems.

A translate of a subgroup, written aH or Ha. Left cosets of a subgroup partition the group into equal-sized blocks, and their number is the index of the subgroup.

A subgroup H of G such that gHg⁻¹ = H for every g in G. Normality is exactly the condition needed for the set of cosets to form a quotient group.

A map between algebraic structures that preserves the operation, so f(ab) = f(a)f(b) for groups. Its kernel measures the failure of injectivity and its image is a subgroup of the codomain.

The group G/H formed by the cosets of a normal subgroup H, with multiplication inherited from G. It captures the structure of G after collapsing H to the identity.

A group generated by a single element, so every element is a power of that generator. Every cyclic group is abelian, and finite ones are isomorphic to integers modulo n.

A set with two operations, addition and multiplication, where addition forms an abelian group and multiplication is associative and distributes over addition. Integers and polynomials are standard examples.

A subgroup of a ring closed under multiplication by any ring element. Ideals are the kernels of ring homomorphisms and allow construction of quotient rings.

A commutative ring in which every nonzero element has a multiplicative inverse. Rational, real, and complex numbers are fields, and finite fields underlie much of coding and cryptography.

An abelian group of vectors over a field, closed under scalar multiplication and satisfying distributive and associative laws. Linear independence, span, and dimension describe its structure.

A set of vectors is linearly independent when the only linear combination equal to zero uses all zero coefficients. Independence guarantees that representations in terms of those vectors are unique.

A linearly independent set of vectors that spans the whole space. Every vector has a unique coordinate representation in a basis, and all bases of a finite-dimensional space share the same size.

The dimension of the column space of a matrix, equal to the number of pivots in row reduction and to the dimension of the image of the associated linear map.

A scalar λ for which some nonzero vector v satisfies Av = λv. Eigenvalues determine stability, diagonalisability, and the long-run behaviour of iterative matrix processes.

A nonzero vector whose direction is unchanged by a linear map, only rescaled by the corresponding eigenvalue. Eigenvectors of distinct eigenvalues are linearly independent.

Writing a matrix as PDP⁻¹ with D diagonal, possible exactly when there is a basis of eigenvectors. It simplifies powers and exponentials of the matrix.

The determinant det(A − λI) of a square matrix, a polynomial whose roots are precisely the eigenvalues. Its degree equals the size of the matrix.

A map between vector spaces preserving addition and scalar multiplication. Every such map between finite-dimensional spaces is represented by a matrix once bases are chosen.

The value a function or sequence approaches as its input or index tends to a point. Limits formalise continuity, derivatives, and convergence of series in real analysis.

A function is continuous at a point when its limit there equals its value at that point. On intervals this means small changes in input produce small changes in output.

The property that a function has a derivative at a point, meaning the difference quotient has a limit. Differentiability implies continuity, but the converse fails.

The derivative of a multivariable function with respect to one variable while holding the others fixed. Partial derivatives assemble into the gradient and appear in optimisation and the chain rule.

The vector of partial derivatives of a scalar function. It points in the direction of steepest increase and is perpendicular to level curves or surfaces.

A series converges when its sequence of partial sums has a finite limit. Standard tests include comparison, ratio, root, and integral tests for positive-term series.

A sequence of functions converges uniformly when the rate of convergence does not depend on the point. It preserves continuity and permits term-by-term integration and, with care, differentiation.

An equation relating an unknown function of one variable to its derivatives. Order and linearity classify such equations, and initial conditions typically select a unique solution.

A function multiplied through a differential equation to make it exact or directly integrable. For first-order linear equations it is exp of the integral of the coefficient of y.

An integral transform converting a function of time into a function of a complex parameter. It turns linear differential equations into algebraic equations and is widely used in engineering.

A measurable function assigning a real number to each outcome of a probability space. Its distribution describes the probabilities of the values it takes.

The function F(x) giving the probability that a random variable is at most x. It is nondecreasing, right-continuous, tends to 0 and 1 at the extremes, and fully determines the distribution.

For a continuous random variable, a nonnegative function whose integral over an interval gives the probability of falling in that interval. It integrates to one over the whole line.

The probability-weighted average value of a random variable, computed by a sum or integral. It is linear, so the expectation of a sum equals the sum of expectations.

The expected squared deviation of a random variable from its mean, measuring spread. It equals the second moment minus the square of the mean and is not additive unless variables are uncorrelated.

The expected product of the deviations of two random variables from their means, measuring joint linear variation. Zero covariance means uncorrelated, which is weaker than independence.

Covariance divided by the product of the two standard deviations, giving a number between minus one and one. It measures the strength and direction of linear association.

The positive square root of the variance, expressed in the same units as the random variable. It is the most common summary of dispersion.

The function E[e^{tX}] of a random variable, whose derivatives at zero give the moments. When it exists in a neighbourhood of zero it uniquely determines the distribution.

States that sums or averages of many independent, identically distributed variables with finite variance approach a normal distribution after standardisation. It justifies normal approximations in statistics.

The parameter value that maximises the likelihood of the observed sample. Under regularity conditions it is consistent and asymptotically normal, though it may be biased in small samples.

A function of the sample that captures all information about the parameter contained in the data. The factorisation criterion characterises sufficiency, and the Rao-Blackwell theorem uses it to improve estimators.

A random interval, computed from data, that contains the true parameter with a stated long-run probability. Its coverage refers to repeated sampling, not to a single realised interval.

A procedure comparing data against a null hypothesis using a test statistic and a rejection region. Errors of the first and second kind trade off through the chosen significance level.