Use the Fundamental Theorem of Homomorphism for Groups to prove the following theorem, which is called the Zassenhaus (Butterfly) Lemma:
Let H and K be subgroups of a group G and H' and K' be normal subgroups of H and K, respectively. Then
i)
ii)
iii)
The situation can be represented by the subgroup diagram below, which explains the name ‘butterfly’.
PART-B (MM: 30 Marks)
(Based on Block 3.)
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Give the smallest for which An is non-abelian. Justify your answer.
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.
न्यूटन-रैफसन विधि द्वारा समीकरण का सन्निकट मूल ज्ञात कीजिए।
लेकर केवल 3 पुनरावृत्तियाँ कीजिए।
ख) द्विघाती समीकरण के मूल
और
दिए गए हैं। दिखाइए कि पुनरावृत्ति
के समीप अभिसरित होगी जब
के मान ज्ञात कीजिए।
Which of the following statements are true, and which are false? Give reasons for your answers.
i) If k is a field, then so is .
ii) If R is an integral domain and I is an ideal of R, then .
iii) In a domain, every prime ideal is a maximal ideal.
iv) If R is a ring with zero divisors, and S is a subring of R, then S has zero divisors.
v) If R is a ring and is of degree
, then f(x) has exactly n roots in R.
Evaluate by using trapezoidal rule with
and
. Use Romber's m
) 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.
निम्नलिखित समीकरण निकाय को हल करने के लिए गाउस–सीडल पुनरावृत्ति विधि को आव्यूह रूप में स्थापित कीजिए :
$
दिखाइए कि पुनरावृत्ति विधि अभिसरित होती है और अतः इसकी अभिसरण दर ज्ञात कीजिए। (5)