IST Groups and Rings (2017)

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Venue: Himachal Pradesh University,  Shimla, Himachal Pradesh
Date:  5th, Jun 2017 to 17th, Jun 2017


Convener(s)
Name R. P. Sharma Dinesh Khurana
Mailing Address Himachal Pradesh University, Shimla.
Panjab University, Chandigarh.

 


 

Speakers and Syllabus 

Speakers:

  1. J. K. Verma, IIT Mumbai
  2. Dinesh Khurana, Panjab University, Chandigarh
  3. S. Parvathi, Ramanujam Institute, Chennai
  4. Chanchal Kumar, IISER Mohali

Course Associates

  1. Rahul Dattatraya, HRI Allahabad
  2. Abhay Soman, IISER Mohali
  3. R. P. Sharma, HPU Shimla
  4. Dinesh Khurana, PU Chandigarh

 

Time Table

  • Tea Breaks: 11.00 - 11.30 and 3.30 - 4.00
  • Lunch Break: 1.00 - 2.30
  • Snacks: 5.00 - 5.30
  9 – 10.30
Lecture 1
(Group Theory)
10.30-11.30 11.30-1.00
Lecture 2
(Group Theory)
2.30-3.30
Tutorial 1
4.00-5.00
Tutorial 2
June 5 JKV Inaugural Function/Tea JKV JKV, RD, DK JKV, RD, DK

 

  9.30 - 11.00
Lecture 1
(Group Theory)
11.30-1.00
Lecture 2
(Ring Theory)
2.30-3.30
Tutorial 1
4.00-5.00
Tutorial 2
June 6 JKV DK DK, RD, JKV DK, RD,JKV
June 7 JKV DK JKV, RD, DK JKV, RD, DK
June 8 JKV DK DK, RD, JKV DK, RD, JKV
June 9 JKV DK DK, RD DK, RD
June 10 DK
(Ring Theory)
DK DK, RD DK, RD
June 12 SP CK SP, AS, RPS SP, AS, RPS
June 13 SP CK CK, AS, DK CK, AS, DK
June 14 SP CK SP, AS, RPS SP, AS, RPS
June 15 SP CK CK, AS, DK CK, AS, DK
June 16 SP CK SP, AS, DK SP, AS, DK
June 17 SP CK CK, AS, DK CK, AS, DK

 

Lectures by Prof. J. K. Verma
(six lectures, each of 1.30 hour)

June 5.  The lectures started with an introduction to group of symmetries of an object and group actions. Cauchy’s theorem; isomorphisms
between certain finite matrix groups and permutation groups and Burnside’s theorem on number of orbits under a group action were discussed.
(Two lectures)
June 6.  The symmetries (isometries) of Euclidean space and orthogonal transformations were introduced, including certain reflections, namely House-
holder transformations. The main theorem in the lecture was Cartan-Dieudonn ́en theorem: every orthogonal transformation of R can be written as product of at
most n Householder reflections. As an application of it, Euler’s theorem on rotations in R3 was proved.
June 8. The lecture started through finite subgroups of rigid motions of Rn , by showing that it suffices to consider finite subgroups of orthogonal groups.
Moving to dimensions n = 3, first certain finite subgroups of rotations in R3 were discussed, which included symmetries of regular polygons and regular polyhedrons. Also, the class equation of group of rotational symmetries of icosahedron and its simplicity was discussed.
June 9. The classification of finite subgroups of SO(3) was the point of discussion on this day. The action of a finite subgroup of SO(3) on a finite set consisting of its poles on unit sphere in R3 was discussed, concluding that all the finite subgroups were of one of the following types: cyclic, dihedral, A4 , S4
and A5.
.Tutorials  Two problem sessions of one hour each on this topic were conducted on June 5, 7 and 10 in the afternoon. The problem sessions were fruitful, the participants were solving problems with a group of two or three people. The
problems were oriented towards a detailed understanding of the topics of discussion on that day or before. The participants enjoyed the study of groups with a geometric viewpoint. The participants raised many good questions during tutorial sessions as well as lectures.
Lectures by Prof. Dinesh Khurana
(six lectures, each of 1.30 hour)
6 June The lecture started with a brief history of commutative ring theory and certain integral domain which arose from study of Fermat’s last theorem.
The main points of discussion were UFD, PID, primes and irreducible elements in integral domains and fundamental theorem of arithmetic. The lecture ended with interesting applications of factorizations to certain problems in number theory: Pythagorean triples and characterizing primes in Z which are sum of two squares.
7 June The lecture started with discussion of UFD’s and non-UFD’s. The algebraic integers in number fields were discussed. The main theorem in the lecture was determination of algebraic integers in quadratic number fields, and
whether they are UFD’s.
8 June The main points  of discussion were Eudlidean domains, the ring of algebraic integers in Q( √d) is a Euclidean domain (hence UFD) if d = −1,
−2, −3, −7, −11, and is UFD if further d = −19, −47, −163 (a conjecture by Gauss). The lecture was ended by Gauss theorem on primitive polynomials and Eisenstein’s irredicubility criteria.
9 June  The main points discussed in the lecture were - The ring of integers in Q(√ −19) is PID but not a Euclidean domain; determination of primes in Z[i]; introduction to Noetherian rings Hilbert basis theorem and Hilbert’s
Nullstellensatz.
10 June (Two lectures) The main theorem in the first lecture on this day was Artin-Tate Lemma. As a consequence the Hilbert’s Nullstellensatz can be obtained as a corollary to the Lemma. The second lecture on this day was
devoted to a detailed understanding of prime and maximal ideal in a polynomials ring R[x] over a PID R.
Tutorial Two problem sessions of one hour each on this topic were conducted on June 6, 8 and 9 after lunch. The problems were selected with a view towards factorizations in domains - unique or non-unique, and also a de-
tailed understanding of the topics under discussion on the corresponding day.Many typical examples of integral domains were discussed. The participants were grouped into two or three during problem solving course, and the choice of problems kept them busy during the whole sessions. Participants raised interesting questions in the lectures as well as problem sessions.
Lectures by Prof. S. Parvathi
12 June In this lecture the vertical Brauer graphs with 2n vertices was introduced. This concept was illustrated with some examples with small n.Participants were asked to draw graphs for n = 3. The group structure was defined on the vertical Brauer graph, and an isomorphism between the group of symmetries on n vertices, Sn and the vertical Brauer graph with 2n vertices was stated. The concepts of Normal subgroup, quotient group, action of group on a set, orbit and stabilizer for the action, etc. were recalled. These concepts were illustrated with concrete examples. The normal subgroup An of Sn does not have a subgroup of order 6 was proven. This was done in connection with converse of the Lagrange’s theorem.
13June  The aim of this lecture was to prove Sylow’s theorems. In order to do this the Isomorphism theorems for groups were first stated and proved. For an action of a p−group on a finite set X, the cardinality of X, and the cardinality
of fixed points of X under the action of p-group are congruent modulo p was proved. After that the proof of the first Sylow’s theorem was given. Sylow’s theorems were applied to classify groups of order 6.
14 June In this lecture Sylow’s second and third theorem were proved.
The notions of internal, and external direct product for groups were introduced.
Sylow’s theorem was used to show groups of order 20,36,48,”and” pq (where p,
and q are prime numbers) are not simple.
15 June  Some more applications of Sylow’s theorem were theme of this lecture. The group of order 255 is cyclic was proved using Sylow’s theorem. For a group G of order pqr (with p < q < r) following was proved:(i) G has a unique subgroup of order r,(ii) G has a normal subgroup of order qr, and (iii) if q does not divide r − 1, then Sylow-q subgroup is normal. Furthermore, the groups of order upto 15 (except groups of order 12 ) were classified
16 June  This lecture started by giving the classification of groups of order 12. The structure theorem for a finite abelian p−group was proved. Also how to find invariant decomposition factors of a given finite abelian p−group was demonstrated. Next, the notion of a subnormal series for a finite group was introduced, and the Jordan-Holder theorem was stated.
17June In this lecture proofs of all the theorems stated in the earlier class was proved. The definition of solvable group was given, and it was illustrated by examples.
Tutorials Two problem sessions of one hour each on this topic were conducted on June 12, 14 and 16 after lunch.
The tutorial problems were designed so that they will illustrate basic notions, and develop computational techniques related to various concepts introduced in lectures. Participants were encouraged to do actual computations for groups of small orders. For instance, participants were asked to use Sylow’s theorem to see a given group is not simple; to compute particular Sylow subgroups and see, by actual computations, that these are indeed conjugate to each other, and finding composition series for a given groups; finding invariant factor decomposition for a given finite abelian p−group, etc. Participants were encouraged to try these problems on their own, and hints were given whenever required.
Lectures by Prof. Chanchal Kumar
12 June  In this lecture definition and some basic properties of algebraic integers were recalled. The ring of algebraic integer for an imaginary quadratic number field was described. The characterisation of prime ideals in Z[ (−5)] was discussed. The motivation was given to learn the theory of imaginary quadratic field, and to understand the ideal class group of a ring of an algebraic integers.
13 June This lecture was mostly devoted to the introduction of lattices in R 2 , and their various properties. The notions of lattice basis, parallelogram
spanned by lattice basis vectors were introduced. The relation between index of a sub-lattice M in a lattice L (respectively, effect of scaling a lattice L by an integer n) in terms of ratio of areas of parallelogram of M , and L (respectively,ratio of areas of parallelogram of L, and nL) was √ given. These notions were illustrated by drawing particular lattices in Z[ (−5)]. An important result that for a ring of an algebraic integers R in an imaginary quadratic number field, the product of a nonzero ideal A in R, and its conjugate A  ̄ is a principal ideal was proved. The notion of divisibility of an ideal in algebraic integers in an imaginary quadratic field was introduced, and illustrated by an example.
14 June In this lecture various important properties of ring of algebraic integers R in an imaginary quadratic number field was proved. Using lattices,it was proved that the index (as an additive group) of a nonzero ideal B in an algebraic integers R is finite, and there are only finitely many ideals of R containing B. Also it was shown that ideal is prime if and only if it is maximal. Furthermore, it was proved that the ring of algebraic integer R for an imaginary quadratic number field is UFD if and only if it is PID. The characterisation of prime ideals in algebraic integers in an imaginary quadratic number field was given. The factorisation of a nonzero ideal into product of primes, and the uniqueness of this factorisation was proved. The notion of a norm for an ideal in a ring of algebraic integers, R in an imaginary quadratic number field was introduced, and various properties of this norm were stated. The notion of ideal class group of R was introduced and this group is finite abelian was stated.
15 June This lecture was devoted to a proof of the result that the ideal class group of a ring of integers in an imaginary quadratic number field is finite, and it is generated by ideal classes of prime ideals P such that the norm of P ,N (P ) is at most μ (the number μ is defined in such a way that it depends only
on the integer d used to define the ring of algebraic integers in an imaginary quadratic number field). The ideal class groups for ring of integers corresponding to −d = 1, 2, 3, 5, 7, 11, 14, 19, 21, 23, 43, 47, 61, 67, 71, 163 were computed. The last part of the lecture was devoted to a brief overview of the theory of algebraic integers in a real quadratic number field.
16 June The aim of this lecture was to discuss Smith normal forms of a m × n matrix over Z. Towards that end, the concept of a module (respectively,
submodule, free module) over a commutative ring with unity; homomorphisms (respectively, isomorphism) of modules; change of basis matrix for linear transformation of free modules over a commutative ring were discussed.
17 June In this lecture the structure theorem for modules over polynomial ring in one variable was proved. The Jordan canonical form for a matrix of a
linear operator was discussed, and illustrated by examples.
Tutorial Two problem sessions of one hour each on this topic were conducted on June 13, 15, and 17.The tutorial problems were so that they will help develop techniques in explicit calculations, and help illustrate finer points in the theory developed. For instance, participants were asked to check whether or not a given rational integer will remain a prime in the given ring of integers, to find explicit factorisation of a nonzero ideal into products of prime, etc. Furthermore, participants were asked to compute the ideal class group of given ring of integers of imaginary quadratic field. Participants were able to compute some of the ideal class groups on their own, and seems to appreciate the theory developed. In the last tutorial session, participants were asked to compute Smith normal form of a given matrix over Z and to do some excercises on free modules over a commutative ring with unity.
Participants were encouraged to try tutorials problems on their own. The hints to solve problems were given whenever required.

 


 

Actual Participants 

 

Sr. SID Full Name Gender Affiliation Position in College/ University University /Institute M.Sc./ M.A.

Year of Passing M.Sc. / M.A

Ph.D. Deg. Date
1 11307 Mr. Parminder Singh Male S.Govt.College Of Science Education & Research Jagraon Asst. Prof. P.U.CHD 2014  
2 11282 Dr. Aditya Mani Mishra Male Rajasthan Technical University Asst. Prof. Motilal Nehru NIT Allahabad 2010 5/13/2014
3 11294 Dr. Dinesh Kumar Male Deen Dayal Upadhyaya College, University of Delhi Asst. Prof. Hindu College, University of Delhi 2009 11/19/2016
4 10109 Ms. Chanpreet Kaur Female Janki Devi Memorial College Asst. Prof. University of Delhi 2010  
5 10969 Ms. Priya Gupta Female Jecrc University Jaipur Rajasthan Asst. Prof. VBS Purvanchal University, Jaunpur 2007 2/11/2015
6 10136 Ms. Teena Kohli Female Janki Devi Memorial College,University of Delhi Asst. Prof. University of Delhi 2008  
7 11044 Dr. Hemant Kalra Male Thapar University, Patiala Asst. Prof. Guru Nanak Dev University, Amritsar 2008 11/14/2013
8 9912 Mr, Aditya Bhan Ojha Male SCVB GOVT. COLLEGE PALAMPUR Asst. Prof. MDS university Ajmer 2009  
9 10386 Mr. Pawanveer Singh Male Lajpat Rai D. A. V. College Jagraon Asst. Prof.  D. A. V. College Jalandhar 2008  
10 10150 Prof. Vijaykumar Jayantibhai Solanki Male Government Engineering College,Bharuch Asst.prof. The M S Uni of Baroda,Vadodara 2009  
11 11324 Mr. Brian Savio Dsouza Male St. Xavier's college (Goa University) Asst. Prof. Goa University 2007  
12 10503 Mr Gaurav Kumar Male Govt. Degree College Hiranagar Asst. Prof. University of Jammu, Jammu 2011  
13 9918 Dr. Nirmal Singh Male NSCBM Govt College Asst. Prof. Kurukshetra University Kurukshetra 2006 11/2/2015
14 11355 Mr. Krishnendu Das Male Netaji Subhas Mahavidyalaya Asst. Prof. University of Hyderabad 2009  
15 10985 Mr. Aryan Kanjibhai Patel Male M.N.College,Visnagar, Gujarat Asst. Prof. Sardar Patel University, Vallabh Vidyanagar 2004  
16 11334 Dr. Dheerendra Mishra Male The Lnm Institute of Information Technology, Jaipur Asst. Prof. M. Sc. 2005 4/3/2014
17 11175 Mr. Heramb Balkrishna Aiya Male Government College Of Arts, Science And Commerce
Quepem - Goa.
Asst. Prof. Goa University 2005  
18 10277 Mr. Amit Sharma Male Pratap Institute Of Technology And Science Asst. Prof. Maharshi Dayanand University, Rohtak 2006  
19 10079 Dr. Avanish Kumar Chaturvedi Male Department Of Mathematics, University Of Allahabad. Asst. Prof. U. P. Autonomous College, Varanasi. 2003 7/25/2009
20 10247 Dr. Gyan Chandra Singh Yadav Male University of Allahabad Asst. Prof. University of Allahabad 2004 5/5/2010
21 11179 Dr. Jeetendra Aggarwal Male Shivaji College, University Of Delhi Asst. Prof. University Of Delhi 2004 10/31/2011
22 10441 Dr. Narinder Sharma Male G.G.M Science College, Canal Road jammu Asst. Prof. University of jammu 2004 10/31/2012
23 10564 Mr. Sanjay Kumar Gupta Male Dev Samaj Post Graduate College For Women Asst. Prof. Panjab University, Campus 1995  
24 10611 Dr. Bibhas Chandra Saha Male Chandidas Mahavidyalaya Asst. Prof. The University of Burdwan 1995 7/2/2010
25 10396 Dr. Madhu Dadhwal Female Himachal Pradesh University
Summer Hil Shimla
Asst. Prof. M.Sc. 2004 10/25/2010
26 10446 Mr Manoj Kumar Male Gautam Group Of Colleges Hamirpur H.P. Asst. Prof. M.S.U. 2010  
27 10315 Dr. Vikram Singh Kapil Male Government Degree College Jukhala Asst. Prof. Himachal Pradesh University 1998 6/6/2008
28 10217 Dr. Tilak Raj Sharma Male H.P.U. Regional Centre Asst. Prof. M.Sc. 1996 8/20/2007

 List of local participants:

Sr.

Name

Gender

Affiliation

Position in College/Uni.

University

M.Sc/M.A.

Year of passing

1.

Sapna

Female

Himachal Pradesh University Summer Hill Shimla-05

Research Scholar

HPU

2011

2.

Virender Sharma

Male

Himachal Pradesh University Summer Hill Shimla-05

Research Scholar

HPU

2011

3.

Nidhi Thakur

Female

Himachal Pradesh University Summer Hill Shimla-05

Research Scholar

HPU

2010

4.

Neetu Dhiman

Female

Himachal Pradesh University Summer Hill Shimla-05

Research Scholar

HPU

2010

5.

Richa Sharma

Female

Himachal Pradesh University Summer Hill Shimla-05

Research Scholar

HPU

2012

6.

Meenakshi

Female

Himachal Pradesh University Summer Hill Shimla-05

Research Scholar

HPU

2012

7.

Arun Kumar

Male

Himachal Pradesh University Summer Hill Shimla-05

Research Scholar

HPU

2012

8.

Pankaj Kumar

Male

Himachal Pradesh University Summer Hill Shimla-05

Research Scholar

HPU

2010

9.

Sumixal Sood

Male

Himachal Pradesh University Summer Hill Shimla-05

Research Scholar

HPU

2014

10

Mudita Sharma 

Female

Delhi University

-

Delhi University

2016

 


How to reach

The Himachal Pradesh University(HPU) Shimla is situated in Summer Hills, Shimla. The airport at Jubar-Hatti is 23 km away. The main Bus Stand is at a distance of 4 km and Railway Station is 4 km. Local buses and taxis are available from railway station and bus stand to Summer Hill.

1. From Delhi, there is direct service of buses (Volvo/AC/Deluxe) from Delhi to Shimla.  Here are some bus sites:

2. Train facility is also available from Delhi to Shimla via Kalka.

3. Some possible train route from Delhi to Shimla is

  • 14095 Himalayan Queen Delhi 05:45- Kalka 11:10
  • 12011 Kalka Shatabdi  Delhi 07:45 - Kalka 11:45
  • 52455 Himalayan Queen Delhi Kalka 12:10 to Shimla 17:20

  
4. One can also travel by train to Kalka and take a number of buses to Shimla. Also the distance  is just 70 km and there are number of taxi available from Kalka to Shimla.