Prove that every non-trivial subgroup of a cyclic group has finite index. Hence prove that is not cyclic.
Let (G) be a cyclic group generated by (g), i.e., (G = \langle g \rangle), and let (H) be a non-trivial subgroup of (G). Since (G) is cyclic, every element of (H) can be written as a power of (g). Let (h) be a non-identity element ________ __________ ___ ________ _______ _____ ______ _______ ____.
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Find all the units of .
Find the approximate root of the equation using Newton-Raphson method. Perform only 3 iterations with
.
The velocity of a vehicle beginning from rest is given in the following table for part of the first four. Using Simpson's rule, find the distance travelled by the vehicle in this hour:
|
| 10 | 20 | 30 | 40 | 50 | 60 |
|
| 80 | 60 | 70 | 75 | 70 | 80 |
बताइए कि निम्नलिखित कथन सत्य हैं या असत्य। अपने उत्तर की पुष्टि के लिए एक लघु उपपत्ति या प्रत्युदाहरण दीजिए।
क) समीकरण का अंतराल [3, 5] में कोई मूल नहीं है।
ख) , जहाँ E स्थानांतरी संकारक और
अग्रांतर संकारक हैं।
ग) प्रत्येक रैखिक समीकरण निकाय LU वियोजन विधि से हल किया जा सकता है।
घ) आंकड़ों (2, 4), (1, 5), (3, 6) के लिए न्यूटन विभाजित अंतर f[x0, x1, x2] का मान है।
ङ) न्यूटन-रैफसन विधि से किसी धन वास्तविक संख्या का घनमूल ज्ञात नहीं किया जा सकता है।
) 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.
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 .
List two distinct cosets of in D10, where r is a reflection in D10.