Using Cayley’s theorem, find the permutation group to which a cyclic group of order 12 is isomorphic.
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) Evaluate by using trapezoidal rule with
and
. Use Romber's method to find the best value of
.
) Estimate the eigenvalues of the matrix
using the Gerschgorin bounds.
Using synthetic division method, show that 2 is a simple root of the equation
.
Which of the following statements are true? Give reasons for your answers.
i) If a group G is isomorphic to one of its proper subgroups, then .
ii) If x and y are elements of a non-abelian group (G, ) such that , then
or
, where e is the identity of G with respect to $$.
iii) There exists a unique non-abelian group of prime order.
iv) If , where A is a group, then
.
v) If H and K are normal subgroups of a group G, then .
Which of the following statements are true, and which are false? Give reasons for your answers.
i) For any ring R and .
ii) Every ring has at least two elements.
iii) If R is a ring with identity and I is an ideal of R, then the identity of R/I is the same as the identity of R.
iv) If is a ring homomorphism, then it is a group homomorphism from (R, +) to (S, +).
v) If R is a ring, then any ring homomorphism from into R is surjective.
Determine a unique polynomial f(x) of degree such that
,
,
,
, where
.
Prove that every non-trivial subgroup of a cyclic group has finite index. Hence prove that is not cyclic.
Set up the Gauss-Seidel iteration scheme in matrix form for solving the system of equations
Show that the method is convergent and hence find its rate of convergence.