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 .
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Using Classical Runge-Kutta fourth order method, find an approximate value of y(1.2) for the IVP ,
with
.
Let S be a set, R a ring and f be a 1-1 mapping of S onto R. Define + and on S by:
.
Show that is a ring isomorphic to R.
PART-C (MM: 20 Marks)
(Based on Block 4.)
Give the smallest for which An is non-abelian. Justify your answer.
) 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.
Prove that every non-trivial subgroup of a cyclic group has finite index. Hence prove that is not cyclic.
Prove that a cyclic group with only one generator can have at most 2 elements.
If , find the values of C1 and C2.