Question

नोट- प्रश्न क्रमांक 01 से 05 तक के प्रश्न लघुउत्तरीय प्रश्न हैं। प्रत्ये
Q.1 Define normal and subnormal series with example. .
Q.2 State the primary decomposition theorem.
Q.3 What is solution of polynomial equation by radicals?
Q.4 State Hilbert basis theorem and Noether lasker theorem.
Q.5 What are the national canonical form and its application?
नोट- प्रश्न क्रमांक 06 से 10 तक के प्रश्न दीर्घउत्तरीय प्रश्न हैं। प्रत्येक
Q.1 State and Prove Jordan - Holder Theorem​

Answer 1

1. If in addition G>Gi for all i, then it is called a normal series. The factors of the series are the quotient groups Gi/Gi+1. ... A subnormal series G = G0 > G1 > ··· > Gn = 〈e〉 is a composition series if each factor Gi/Gi+1 is simple. It is a solvable series if each factor is abelian.

2.The primary decomposition theorem allows us to use the minimal polynomial of a matrix to write a vector space as a direct sum of invariant subspaces. It is key to understanding and interpreting the information provided by the minimal polynomial.

3.Solving a polynomial by radicals is the expression of all roots of a polynomial using only the four basic operations: addition, subtraction, multiplication and division, as well as the taking of radicals, on the arithmetical combinations of coefficients of any given polynomial.

4.Hilbert's Basis Theorem. If {\displaystyle R}R is a Noetherian ring, then {\displaystyle R[X]}R[X] is a Noetherian ring.

5.In most fields, a canonical form specifies a unique representation for every object, while a normal form simply specifies its form, without the requirement of uniqueness. The canonical form of a positive integer in decimal representation is a finite sequence of digits that does not begin with zero.

1..State and Prove Jordan - Holder Theorem--------------_________==== The Jordan–Hölder theorem is a theorem about composition series of finite groups. A composition series is a chain of subgroups

1 = H_0 \triangleleft H_1 \triangleleft H_2 \triangleleft \cdots \triangleleft H_{k-1} \triangleleft H_k = G,

1=H

0

◃H

1

◃H

2

◃⋯◃H

k−1

◃H

k

=G,

where H_iH

i

is a maximal proper normal subgroup of H_{i+1}.H

i+1

. By the third isomorphism theorem, this is equivalent to the statement that the quotient H_{i+1}/H_iH

i+1

/H

i

is a simple group. This quotient is called a composition factor

Step-by-step explanation:

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