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)	10.30-11.30	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 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 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.