Online TA Profiles
Thokchom Sarojkumar Sinha, MSc
OTA ID#: 104119

Education Experience: BSc, Mathematics, Physics, Chemistry, English, Manipur University, 1985
MSc, Mathematics, Kanpur University, 1996
Focus of Study: Although I plan on completing my PhD, for the time being I am not working on it.
Work Experience: I have been serving as a Graduate Teacher since 1987 in Manipur State, India.
Skills & Achievements: I have 15 years of teaching experience in mathematics.
Career Interests: I'd like to become a lecturer in college or in university.
Outside Interests: I like reading science magazines and other novels.
Message to Students
and/or Parents:		 I'd for students to pay proper attention to their studies.
Postings Answered: 619
Cumulative OTA Rating: 4.7/5  What is OTA Rating?
Top Solutions Downloads
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  1. Abelian group - Modern Algebra Group Theory (LIX) Quotient Group or Factor Group ...
  2. Let F be the field of real numbers and let V be the set of all sequences ( a1, a2, … , an, … ), ai Є F, where equality, addition and scalar multiplication are defined componentwise. Then V is a vector space over F. Let W = {( a1, a2, … , an, … ) Є V│limn→∞an = 0 }. Then W is a subspace of V. Let U = {( a1, a2, … , an, … ) Є V│∑ai2 is finite, where i is varying from 1 to infinity }. Prove that U is a subspace of V and is contained in W, where W = {( a1, a2, … , an, … ) Є V│limn→∞an = 0 }. - Modern Algebra Vector Spaces and Modules (VI) Subspaces of a Vector Space ...
  3. Group Theory : Conjugates and Permutation Groups - Find the number of conjugates that the r-cycle (1 , 2 , … , r) has in Sn .
  4. Theory of Numbers : Fibonacci Numbers - Prove that (Fn+1)^2 - Fn Fn+2 = (- 1)^n
  5. Group Theory : Let G be a group, H a subgroup of G, T an automorphism of G. Let (H)T = {hT| hєH}.Prove (H)T is a subgroup of G. - Let G be a group, H a subgroup of G, T an automorphism of G. Let (H)T = {hT|hєH }.Prove (H)T is a subgroup of G.
  6. Group Theory : If in a finite group G an element a has exactly two conjugates, prove that G has a normal subgroup N ≠ e , G. - If in a finite group G an element a has exactly two conjugates, prove that G has a normal subgroup N ≠ e , G
  7. Group Theory : Find the number of conjugates of (1 2)(3 4) in Sn , n ≥ 4. - Find the number of conjugates of (1 2)(3 4) in Sn , n ≥ 4.
  8. Group Theory : If G is a group and H is a subgroup of index 2 in G, prove that H is a normal subgroup of G. - If G is a group and H is a subgroup of index 2 in G, prove that H is a normal subgroup of G.
  9. Group Theory If H is a subgroup of G, let N(H) = {gєG|gHg^-1 = H}. Prove that if H is a normal subgroup of the subgroup K in G, then K is subset N(H)( that is, N(H) is the largest subgroup of G in which H is normal). - If H is a subgroup of G, let N(H) = {gєG|gHg^-1 = H}. Prove that if H is a normal subgroup of the subgroup K in G, then K is subset N(H)( that is, N(H) is the largest subgroup of G in wh ...
  10. Group Theory : If G is a finite group, show that there exists a positive integer N such that a^N=e for all aЄG. - If G is a finite group, show that there exists a positive integer N such that a^N=e for all aЄG.
  11. It is an explanation for solving Non- Homogeneous Linear Partial Differential Equation with Constant Coefficients. Find the solution of the equation (D2 - D'2 + D - D')z = e^(2x + 2y). - Linear Partial Differential Equation (II) Non- Homogeneous Linear Partial Differential Equation with Constant Coefficients ...
  12. Let V be a region in 3complying with the hypotheses of the divergence theorem, and denote by S its boundary surface. Let also φ: →  be a scalar function, and c an arbitrary constant vector. By applying the divergence theorem to the vector field φc (1) show that: (∫∫∫v ▼φdV - ∫∫s φndS).c = 0 with the understanding that the integral of a vector is the vector of the integrals of the components. (2) Use the above result to deduce carefully that: ∫∫∫v ▼φdV = ∫∫s φndS. - Real Analysis Divergence Theorem Let V be a region in  ...
  13. Group Theory 1. i. State the axioms for an equivalence relation ii. The relation n mod 3 divides the non-negative integers (i.e, n in Z such that n ≥ 0) into how many partitions? Show that n = 0 mod 3 is an equivalence relation. 2. Prove that, for any matrices, A, B and C: A+B=B+A And: A+(B+C)=(A+B)+C ( i.e., that the matrix addition is both commutative and associative) For simplicity, prove these properties using 2x2 matrices. 3. Prove that addition modulo n, written + is: i. Associative. ii. Commutative. There are two ways to prove these properties. Each way requires a definition or two: i. For n ≥ 2, 0 ≤ a, b ≤ n+1, a+ b= a+b if a+b< n a+n-n if a+b≥ n ii. Writing a for a mod n and (a+b) = a+ b, then: (p+ q) ≡ (p +q ) Do the proof using both methods. Which is more "algebraic" (in the sense of "abstract" algebra)? 4. Prove that addition modulo n, written + is: i. Associative ii Commutative. ( extra definations required : a for a mod n and (pà-q) = pà- q, so (pà- q) (p à-q ) 5. i. State the axioms defining a group - If (Z, +) is a group, show that it is. Identify the identity element for +; in addition, for each n in Z , identify the inverse. Also show that + is associative. - If (Z, à-) is a group, show that it is. Identify the identity element for +; in addition, for each n in Z, identify the inverse. Also show that + is associative. - If (R, à-) is a group, show that it is. Identify the identity element for +; in addition, for each n in Z , identify the inverse. Also show that + is associative. iii. In each case, deterimine whether the algebraic structure is a group. For each such group: - show how it satifies the group axioms - Draw the cayley table for the group and list the inverse elements i. For S=(0,2), a+2b ≡ (a+b) mod 2 and aà-2b ≡ (a à-b)mod2 a. (S,+ ) ( possibly an additive group) b. (S,∙ 2) (possibly a multiplicative group). ii. For S = (0,1,2) where n=2,3 and + and à- are defined as in the last part. a. (S,+n) ( possibly an additive group) b. (S,∙ n) (possibly a multiplicative group). Determine whether any of the groups is an abelian group. If any of them are abelian: i. state the conditions under which a group is abelian ii. show that the group is abelian 6. there are only two groups of order four (Z 4and v). How many groups are there of the order five? Draw cayley tables for each one of them( the element should be named a,b,c,d,e) is either (or both) of the groups of order four subgroup of any of those of order five? If so which one 7. for each of the following structures, state whethere it is a group. If it is, state whether it is abelian or not. i.For any set, A, the set of one-to one and onto functions, f: A →A under composition ( written "◦"). ii.The set of all subsets of the three-element set (a,b,c) ( there are eight such subsets) under: a. Union b. Intersection iii. The set G=(a+b√5| a,b in Q) under addition and multiplication iv The set consisting of non-zero numbers under a. addition b. division v. The set (1,5,7,11) under multiplication modulo 12. Draw cayley table vi. The set (4,6) under multiplication modulo 12. draw cayley table vii. The set of real numbers under à-, where aà-b = 2(a+b) viii The set of real numbers under +, where a+b = a+b-10 ix. The set of rotational symmetries of a regular hexagon under composition x The following sets of permutations under composition i. (e,(12),(123),(1234)) ii. (e,(12), (34), (12), (34)) 8.Let G be a group, (G, * ) in which there is an element, a , such that g * g=g . prove that g=e 9.Prove that for every element, a, of a group, G, the order of a and a^-1 are the same ( including the case of an infinite order) 10.Let x and y be elements of a group, G. Prove that the elements xy and yx have the same orders 11.Find the subgroups of i. Z7 ii. Z8 iii Z9 12. i. determine which of the folowign are subgroups of under + a. (0) b. (-1,0,1) c. (n| n=10m for some integer m d.(p| p is a prime number e. (0,1,2,3,4) under addition modulo 5 ii. Determine which of the following are subgroups of under mulitiplication: a. (1, -1) b. (x |x=3, for some integer n c. (x |x=p/2ⁿ for some integers, p,n) d. (x| x=k 3 for some interger k - Group Theory ...
  14. Group Theory : Permutation Groups - Prove that the smallest subgroup of Sn containing ( 1 , 2 )and ( 1 , 2 , … , n ) is Sn. ( In other words, these generate Sn )
  15. Group Theory : Find two elements in A5 , alternating group of degree 5 , which are conjugate in S5 but not in A5 . - Find two elements in A5 , alternating group of degree 5 , which are conjugate in S5 but not in A5 .
  16. Abelian group - Modern Algebra Group Theory (X) In a group G in which (a.b)^i =a^i.b^i for three consecutive integers ...
  17. Group Theory (XLIV): Suppose H is the only subgroup of order O(H) in the finite group G. Prove that H is a normal subgroup of G. - Modern Algebra Group Theory (XLIV) Subgroups of a Group Normal S ...
  18. Ring Theory : Euclidean Ring: In a Euclidean ring, prove that any two greatest common divisors of a and b are associates. - In a Euclidean ring, prove that any two greatest common divisors of a and b are associates.
  19. Karnaugh map or K-map:It is an explanation for simplifying Boolean function by using Three-variable maps. Simplify the following Boolean function using Three-variable maps: F(x,y,z) = ∑(1,2,3,6,7) - Computer Organization Digital Logic Circuits(IX) ...
  20. The Order of an Element of a group: If in the group G, a^5 = e, aba^(-1) = b^2 for a,bєG find O(b). - Modern Algebra Group Theory (XXXIII) Subgroups of a Group ...
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