Show that, if G is a finite group and a ∈ G, o(a) | o(G). Further, show that a°(G) = e for all a ∈ G. Deduce the Euler-Fermat theorem a(n) = 1 (mod n) for all a, n ∈ N, n ≥ 2, (a, n) = 1.
See Answer →Calculate the following:
(i) (3x² + 4x + 1) + (3x3 + 4x² + 2x + 3) in Z5[x].
(ii) (3x²+2x+6). (3x3+4x+5) in Z7[x].
See Answer →Let R be a ring in which a² = a for all a ∈ R. Show that a = -a and R is commutative.
See Answer →Let R = Z20
See Answer →If F is a field, show that U(F[x]) = F.
See Answer →Let a = (125), β = (1432) ∈ Ss. Compute σ = α β-¹. Write as a product of transpositons. What is the signature of σ?
See Answer →Let A be a 3 x 4 real matrix, B be a 4 × 2 real matrix and C be a 2 x 3 real matrix. Which of the following operations are defined?
(i) CA + B¹
(ii) AB + C¹
For those operations that are defined, what is the order of the resulting matrix?
See Answer →
Let S = {1, 2, 3, 4} and ∗ be the binary operation defined by a ∗ b = a. Compute the Cayley table for (S, ∗). Is ∗ commutative? Is ∗ associative? Justify your answers.
See Answer →State Lagrange’s theorem. What are the possible orders of subgroups of a group of order 12?
See Answer →Define a subgroup of a group. Check whether
is a subgroup of the the group of 2 × 3 matrices over ℂ under addition
See Answer →Define an abelian group. Give an example of a non-abelian group. (You don’t need to prove that your example is a group. You have to only prove that it is non-abelian.
See Answer →पारिभाषिक शब्दावली की नियोजित विकास प्रक्रिया को स्पष्ट कीजिए।
See Answer →कविता के अनुवाद की प्रमुख चुनौतियों पर सोदाहरण प्रकाश डालिए।
See Answer →नाइडा द्वारा बताए गए अनुवाद के विभिन्न सोपानों की चर्चा कीजिए।
See Answer →रूपांतरण और छायानुवाद में अंतर है? समझाइए।
See Answer →