**S.No.** |
**Content of the topics** |
**Learning Objectives** |
**Teaching Guidelines** |
**Methodology** |
**Time (Hours)** |

1 |
1.Definition of a group with example and simple properties of groups |
Student will be able to understand the concept groups, subgroups and normal groups |
Lecture should be effective so that student will be able to grasp the topics easily |
Assignments/Seminars/ Class Tests/ Presentations |
10 |

2.Subgroups and subgroup criteria |

Generation of groups, cyclic groups |

3.Cosets, Left and right cosets |

4.Index of a sub-group, Coset decomposition |

5.Lagrange’s theorem and its consequences |

6.Normal subgroups, Quotient groups |

2 |
1.Homomorphisms, isomorphisms |
Student will be able to understand homomorphism |
Lecture should be effective so that student will be able to grasp the topics easily |
Assignments/Seminars/ Class Tests/ Presentations |
15 |

2.Automorphisms and inner automorphisms of a group |

3.Automorphisms of cyclic groups |

4.Permutations groups, even and odd permutations |

5.Alternating groups |

6.Cayley’s theorem |

7.Centre of a group and derived group of a group |

3 |
1.Introduction to rings, subrings |
Student will be able to understand study of rings and ideals |
Lecture should be effective so that student will be able to grasp the topics easily |
Assignments/Seminars/ Class Tests/ Presentations |
15 |

2.Integral domains and fields |

3.Characteristics of a ring, Ring homomorphisms |

4.Ideals(principle, prime and Maximal) and Quotient rings |

5.Field of quotients of an integral domain |

4 |
1.Euclidean rings |
Student will study about the polynomial rings |
Lecture should be effective so that student will be able to grasp the topics easily |
Assignments/Seminars/ Class Tests/ Presentations |
12 |

2.Polynomial rings |

3.Polynomials over the rational field |

4.The Einstein’s criterion |

5.Polynomial rings over commutative rings |

6.Unique factorization domain |

7.R unique factorization domain implies so is R[X1,X2,…XN] |