Mathematics (Paper-I ) Group and Ring theory & Linear Algebra ( समूह और वलय सिद्धांत एवं रैखिक बीजगणित ) B.Sc 5th Sem U.P

Unit

Topic

Total No.

of Lectures

Part-A: Group and Ring Theory

Hkkx A& ¼lewg rFkk oy; fl)kar½

I

Introduction to Indian ancient Mathematics and Mathematicians should be included under Continuous Internal Evaluation (CIE). Automorphism, inner automorphism, Automorphism groups, Automorphism groups of finite and infinite cyclic groups, Characteristic subgroups, Commutator subgroup and its properties; Applications of factor groups to automorphism groups. ¼lrr vkarfjd ewY;kadu ¼lhvkbZbZ½ ds varxZr Hkkjrh; çkphu xf.kr ,oa xf.krKksa dk ifjp; 'kkfey fd;k tkuk pkfg,A LoÁfr:i.k] vkarfjd LoÁfr:i.k] LoÁfr:i.k lewg] ifjfer vkSj vifjfer pØh; lewgksa ds LoÁfr:i.k lewg] fo'ks"krk milewg] Øefofue;d milewg vkSj blds xq.k; LoÁfr:i.k lewgksa ds fy, dkjd lewgksa dk vuqç;ksxA½

10

II

Conjugacy classes, The class equation, p-groups, The Sylow theorems and consequences, Applications of Sylow theorems; Finite simple groups, Nonsimplicity tests; Generalized Cayley’s theorem, Index theorem, Embedding theorem and applications. ¼la;qXeu oxZ] oxZ lehdj.k] p &lewg] lkbyks çes; vkSj ifj.kke] lkbyks çes; ds vuqç;ksx; ifjfer ljy lewg] vljyrk ijh{k.k; lkekU;h—r dsyh çes;] lwpdkad çes;] ,acsfMax çes; vkSj vuqç;ksxA½

10

III

Polynomial rings over commutative rings, Division algorithm and consequences, Principal ideal domains, Factorization of polynomials, Reducibility tests, Irreducibility tests, Eisenstein criterion, Unique factorization in Z [x]. ¼Øefofues; oy; ij cgqin oy;] foHkktu ,YxksfjFe vkSj ifj.kke] eq[; xq.ktkoyh izkar] cgqin dk xq.ku[kaMu] y?kqdj.kh; ijh{k.k] vy?kqdj.kh;  ijh{k.k] vkablULVhu ekunaM] Z [x]] esa vf}rh; xq.ku[kaMuA½

9

IV

Divisibility in integral domains, Irreducibles, Primes, Unique factorization domains, Euclidean domains. ¼iw.kkZadh; izkar esa foHkkT;rk] vy?kqdj.kd] vHkkT;] vf}rh; xq.ku[kaMu izkar] ;wfDyfM;u izkarA½

9

Part-B: Linear Algebra

Hkkx-B& ¼jSf[kd chtxf.kr½

V

Vector spaces, Subspaces, Linear independence and dependence of vectors, Basis and Dimension, Quotient space. ¼lfn'k lef"V] milef"V] jSf[kd Lora=rk vkSj lfn'kksa dh fuHkZjrk] vk/kkj vkSj vk;ke] HkkxQy lef"VA½

10

VI

Linear transformations, The Algebra of linear transformations, rank nullity theorem, their representation as matrices. ¼jSf[kd :ikarj.k] jSf[kd :ikarj.kksa dk chtxf.kr] jSad 'kwU;rk çes;] vkO;wg ds :i esa mudk fu:i.kA½

9

VII

Linear functionals, Dual space, Characteristic values, Cayley Hamilton Theorem.

¼jSf[kd Qyud] }Sr Lkef"V] vfHky{kf.kd eku] dsyh gSfeYVu çes;A½

9

VIII

Inner product spaces and norms, Cauchy-Schwarz inequality, Orthogonal vectors, Orthonormal sets and bases, Bessel’s inequality for finite dimensional spaces, Gram-Schmidt orthogonalization process, Bilinear and Quadratic forms. ¼vkarjxq.kuLkef"V vkSj ekunaM] d‚ph&'oktZ vlekurk] v‚FkksZxksuy lfn’k] v‚FkksZu‚eZy leqPp; vkSj vk/kkj] ifjfer vk;keh Lkef"V;ksa ds fy, cslsy dh vlekurk] xzke&f'eV v‚FkksZxksuykbts'ku çfØ;k] f}jSf[kd vkSj f}?kkr :iA½

8