IST Groups and Rings (2017)
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:
 J. K. Verma, IIT Mumbai
 Dinesh Khurana, Panjab University, Chandigarh
 S. Parvathi, Ramanujam Institute, Chennai
 Chanchal Kumar, IISER Mohali
Course Associates
 Rahul Dattatraya, HRI Allahabad
 Abhay Soman, IISER Mohali
 R. P. Sharma, HPU Shimla
 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) 
10.3011.30  11.301.00 Lecture 2 (Group Theory) 
2.303.30 Tutorial 1 
4.005.00 Tutorial 2 

June 5  JKV  Inaugural Function/Tea  JKV  JKV, RD, DK  JKV, RD, DK 
9.30  11.00 Lecture 1 (Group Theory) 
11.301.00 Lecture 2 (Ring Theory) 
2.303.30 Tutorial 1 
4.005.00 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 

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 CartanDieudonn ́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 R^{3} was proved. 
June 8.  The lecture started through finite subgroups of rigid motions of R^{n} , by showing that it suffices to consider finite subgroups of orthogonal groups. Moving to dimensions n = 3, first certain finite subgroups of rotations in R^{3} 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 R^{3} was discussed, concluding that all the finite subgroups were of one of the following types: cyclic, dihedral, A_{4} , S_{4} and A_{5}. 
.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 nonUFD’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 ArtinTate 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 nonunique, 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 2_{n} 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, S_{n} and the vertical Brauer graph with 2_{n} 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 pgroup 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 Sylowq 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 JordanHolder 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 sublattice 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 Shimla05 
Research Scholar 
HPU 
2011 
2. 
Virender Sharma 
Male 
Himachal Pradesh University Summer Hill Shimla05 
Research Scholar 
HPU 
2011 
3. 
Nidhi Thakur 
Female 
Himachal Pradesh University Summer Hill Shimla05 
Research Scholar 
HPU 
2010 
4. 
Neetu Dhiman 
Female 
Himachal Pradesh University Summer Hill Shimla05 
Research Scholar 
HPU 
2010 
5. 
Richa Sharma 
Female 
Himachal Pradesh University Summer Hill Shimla05 
Research Scholar 
HPU 
2012 
6. 
Meenakshi 
Female 
Himachal Pradesh University Summer Hill Shimla05 
Research Scholar 
HPU 
2012 
7. 
Arun Kumar 
Male 
Himachal Pradesh University Summer Hill Shimla05 
Research Scholar 
HPU 
2012 
8. 
Pankaj Kumar 
Male 
Himachal Pradesh University Summer Hill Shimla05 
Research Scholar 
HPU 
2010 
9. 
Sumixal Sood 
Male 
Himachal Pradesh University Summer Hill Shimla05 
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 JubarHatti 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.