Derive the class equation for finite group.

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Algebraic Structures (SMAM11)

1. (A) Derive the class equation for finite group.
   (OR)
   (B) Let G be a group of order 715. Then the sylow 13-subgroup H of G is in Z(G).
2. (A) Let V be an n-dimensional vector space over a field F. Then, given any element T ∈ A(V), there exists a non trivial polynomial q(x) ∈ F[x] of degree at most n², such that q(T) = 0.
   (OR)
   (B) Prove that there exists a subspace W of V, invariant under T, such that V = V₁ ⊕ W.

Real Analysis – I (SMAM12)

1. (A) Let f be bounded variation on [a, b] and assume that c ∈ (a, b). Then f is of bounded variation on [a, c] and on [c, b] and we have V_f(a, b) = V_f(a, c) + V_f(c, b).
   (OR)
   (B) State and prove Euler’s Summation Formula.
2. (A) State and prove Mertens Formula.
   (OR)
   (B) Assume that f_n → f uniformly on S. If each f_n is continuous at a point ‘c’ of S, then the limit function f is also continuous at c.

Ordinary Differential Equations (SMAM13)

1. (A) Find the solution Φ of the initial value problem y”’ + y = 0, y(0) = 0, y'(0) = 1, y”(0) = 0.
   (OR)
   (B) Verify that the function Φ₁(x) = x satisfies the equation x²y”’ – 3x²y” + 6xy’ – 6y = 0, for x > 0. Find the second solution Φ₂. Also show that {Φ₁, Φ₂} form a basis for the solution for x > 0.
2. (A) Find two linearly independent power series solutions (in powers of x) of the equation y” – xy’ + y = 0.
   (OR)
   (B) (i) Solve (6x – 4y + 1)dy = (3x – 2y + 1)dx.
   (ii) Solve cos x cos y dx – 2 sin x sin y dy = 0.

Graph Theory and Applications (SMAE11)

1. (A) An edge e is a cut edge of a connected graph G if and only if there exists vertices u and v such that e belongs to every (u, v) path.
   (OR)
   (B) If G is a graph with v-1 vertices, prove that the following are equivalent.
   (a) G is connected.
   (b) G is acyclic.
   (c) G is a tree.
2. (A) State and Prove Cayley’s recursive formula.
   (OR)
   (B) State and Prove Tutte’s perfect Matching Theorem.

Fuzzy Sets and their Applications (SMAE12)

1. (A) State and Prove Decomposition theorem.
   (OR)
   (B) If R is transitive and reflexive (that is, is a preorder), then R^k = R, k = 1, 2, 3, ….
2. (A) State and Prove theorem of decomposition for a similitude relation.
   (OR)
   (B) State and Prove decomposition theorem for a fuzzy perfect order relation.