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18.09.21

SAT

1.ALGEBRA

UNIT TEST-1

 

ALGEBRA

SYLOW’S THEOREM

          If p is prime number and  , then G has a subgroup of order

 

 

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19.09.21

SUN

1.ALGEBRA

UNIT TEST-2

 

 

ALGEBRA

Second Part of Sylow’s theorem:

If G is finite group , p a prime and  but  ,

 then any two subgroups of G of order pn are conjugate.

 

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20.09.21

MON

2.REAL  ANALYSIS

UNIT TEST-3

 

 

ALGEBRA

Third Part of Sylow’s theorem:

 

                The number of p-sylow subgroups in G, for a given prime, is of the form 1+kp

 

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21.09.21

TUE

2.REAL  ANALYSIS

UNIT TEST-4

 

 

ALGEBRA

         If  , p and q distinct primes, p < q

        (a)   if , then G is cyclic.   

(b)   if ,

then there exists a unique non abelian group order pq.                 https://yakacenter.blogspot.com

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23.09.21

THUR

3.FOURIER SERIES AND FOURIER INTEGRALS

UNIT TEST-5

 

ALGEBRA

CAUCHY CONDENSATION TEST :

      Let G be a group of order pqr, p<q<r primes. Then

        (a)   the r-sylow subgroup is normal in G.

        (b)   G has a normal subgroup of order qr.

        (c)    if  , the q-sylow subgroup of G is normal in G.         https://yakacenter.blogspot.com

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25.09.21

SAT

3.FOURIER SERIES AND FOURIER INTEGRALS

UNIT TEST-6

 

 

ALGEBRA

If G is of order p2,q,p,q primes, then G has a nontrivial normal subgroup.

If G is of order p2,q,p,q primes, then either a p-sylow subgroup or a q-sylow subgroup of G

must be normal in G.

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26.09.21

SUN

4.DIFFERENTIAL GEOMETRY

UNIT TEST-7

 

 

ALGEBRA

Let G be a finite abelian group. Then G is isomorphic to the direct product of

its Sylow subgroups.

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28.09.21

TUE

4.DIFFERENTIAL GEOMETRY

UNIT TEST-8

 

 

ALGEBRA

Let A,B be cyclic groups of order m and n, respectively.

 Then A x B is cyclic if and only if m and n are relatively prime.

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30.09.21

THUR

5.OPERATIONS RESEARCH

UNIT TEST-9

 

 

ALGEBRA

Every group of p2,p a prime number, is either cyclic or is isomorphic to the direct product of two cyclic groups each of order p.

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02.10..21

SAT

5.OPERATIONS RESEARCH

UNIT TEST-10

 

 

ALGEBRA

Two abelian groups of order pn are isomorphic if and only if  they have the same invariants.

 

The number of non-isomorphic abelian groups of order pn ,  p a prime,

equals the number of partitions of n.

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03.10.21

SUN

6.FUNCTIONAL ANALYSIS

UNIT TEST-11

 

 

ALGEBRA

 If p is odd prime and if , where a and b are integers,

 then . If , .

In the integers  a prime , there are at most n solutions  for every integer n.

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05.10.21

TUE

6.FUNCTIONAL ANALYSIS

UNIT TEST-12

 

 

ALGEBRA

A group G is said to be solvable if there exist subgroups

G=  such that is normal in  and | is abelian.

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07.10.21

THUR

7. COMPLEX ANALYSIS

UNIT TEST-13

 

 

ALGEBRA

Any subgroup and homomorphic image of a nilpotent group must be nilpotent.

 

Every homomorphic image, different from (e), of a nilpotent group has a nontrivial center

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09.10.21

SAT

COMPLEX ANALYSIS

UNIT TEST-14

 

 

ALGEBRA

               If G is a finite group, then G nilpotent if and only if  G is the direct product of its

Sylow’s subgroups.

If G is a finite group and P is a p-sylow subgroup of G,

  then the number of conjugates of P in G is not a multiple of p.

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10.10.21

SUN

8. DIFFERENTIAL EQUATIONS

UNIT TEST-15

 

 

ALGEBRA

v  A group G is the set-theoretic union of three proper subgroups if and only if G has ,

as a homomorphic image, a noncyclic group of order 4.

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12.05.21

TUE

8. DIFFERENTIAL EQUATIONS

UNIT TEST-16

 

 

ALGEBRA

Gauss’s Lemma:

If the primitive polynomials f(x) can be factored as the product of two polynomials having rational coefficients, it can be factored as the product of two polynomials having integer coefficients.

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14.10.21

THUR

9. STATISTICS –I

UNIT TEST-17

 

 

ALGEBRA

The Eisenstein Criterion

Let f(x)= a0+a1x+……..+anxn be a polynomial with integer coefficients.

 Suppose that for some prime number .

Then f(x) is irreducible over the rational.

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15.10.21

FRI

9. STATISTICS –I

UNIT TEST-18

 

 

ALGEBRA

The polynomial f(x)=1+x+x3+x4  is not irreducible over any field F.

 

The polynomial f(x)=x4+2x+2 is irreducible over the field of rational numbers.

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16.10.21

SAT

10. STATISTICS –II

UNIT TEST-19

 

 

ALGEBRA

                If L is a finite extension of K and if K is a finite extension of F,

then L is finite extension of F.

 Moreover, [L:F]=[L:K][K:F].

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17.10.21

SUN

10. STATISTICS –II

UNIT TEST-20

 

 

ALGEBRA

 

                    If L is an algebraic extension of K and if K is an algebraic extension of F,

then L is an algebraic extension of F.

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19.10.21

TUE

ALL UNITS

REVISION TEST-1

 

 

ALGEBRA

 

                A complex number is said to be an algebraic number if it is algebraic over the field of rational number.

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21.10.21

THUR

ALL UNITS

REVISION TEST-2

 

 

ALGEBRA

 

If p(x) is a polynomial in F[x] of degree and is irreducible over F,

then there is an extension E of F, such that [E:F]=n, in which p(x) has a root.

 

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23.10.21

SAT

ALL UNITS

REVISION TEST-3

 

 

ALGEBRA

 

                                If p is a prime number, then the splitting field over F, the field of rational numbers, of the polynomial  is degree of p-1.

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24.10.21

SUN

ALL UNITS

REVISION TEST-4

 

 

ALGEBRA

          If  is irreducible, then

                 (i) If the characteristic of F is 0, f(x) has no multiple.  

                 (ii) If the characteristic of F is

            has a multiple root only if it is of the form .

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26.10.21

TUE

ALL UNITS

REVISION TEST-5

 

 

ALGEBRA

 

                If K is finite, separable extension of F then K is a simple extension of F.

 

                If one of a or b is separable over F, then F(a,b) is simple extension of F.

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28.10.21

THUR

ALL UNITS

REVISION TEST-6

 

 

ALGEBRA

 

If K is a normal extension of F if and only if K is the splitting field of some polynomial over F.

 

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30.10.21

SAT

ALL UNITS

REVISION TEST-7

 

 

ALGEBRA

Any automorphisms of K must leave every rational number fixed.

 

If the field F contains a primitive nth root of unity, then the Galois group of xn - a for , is abelian.

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31.10.21

SUN

ALL UNITS

REVISION TEST-8

 

 

ALGEBRA

Let F be a finite field then F has pm elements where the prime number p is the characteristic of F.

If the finite field F has pm elements then every  satisfies

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02.11.21

TUE

ALL UNITS

REVISION TEST-9

 

 

ALGEBRA

 

Let G be a finite abelian group enjoying the property that the relation xn=e is satisfied by at most n elements of G, for every integer n. Then G is a cyclic group.

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06.11.21

SAT

ALL UNITS

REVISION TEST-10

 

 

ALGEBRA

                     Let D   be a division ring such that for every  there exists a positive integer  

depending on a, such that . Then D is a commutative field.

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07.11.21

SUN

ALL UNITS

MODEL EXAM-1

 

 

ALGEBRA

The polynomials  form a basis V and hence the dimension of V  is (n+1).

So, the independent set  must also be a basis for V .Thus for each f in V .

This is called Lagrange’s interpolation formula.

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09.12.21

TUE

ALL UNITS

MODEL EXAM-2

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ALGEBRA

Let F be a field. A polynomial f in F[x] is said to be reducible over F

if there exist polynomials g,h in F[x] of  such that  and

 if not, f is said to be irreducible over F.                https://yakacenter.blogspot.com

 

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