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Semester 5: B.Sc. Mathematics
Groups, subgroups and cyclic groups
Groups, subgroups and cyclic groups
Definition of Groups
A group is a set equipped with a binary operation that satisfies four fundamental properties: closure, associativity, identity, and invertibility. For a set G and a binary operation * to form a group, the following must hold: 1. Closure: For all a, b in G, a * b is also in G. 2. Associativity: For all a, b, c in G, (a * b) * c = a * (b * c). 3. Identity: There exists an element e in G such that for all a in G, e * a = a * e = a. 4. Invertibility: For each a in G, there exists an element b in G such that a * b = b * a = e.
Types of Groups
Groups can be classified into several types including: 1. Finite and Infinite Groups: Finite groups have a limited number of elements, while infinite groups do not. 2. Abelian Groups: A group is abelian if it satisfies commutativity, i.e., a * b = b * a for all a, b in G. 3. Non-Abelian Groups: In these groups, commutativity does not hold.
Subgroups
A subgroup is a subset of a group that itself forms a group under the same operation. If H is a subgroup of G, then H must satisfy the group properties as defined for G. Criteria for H to be a subgroup include: 1. The identity element of G is in H. 2. For any a, b in H, the result of the group operation a * b is also in H. 3. For every element a in H, the inverse of a is also in H.
Cyclic Groups
A cyclic group is a group that can be generated by a single element. If G is a cyclic group generated by an element g, then G consists of all integer powers of g, denoted by {g^n | n ∈ Z}. Notably: 1. Every cyclic group is abelian. 2. A finite cyclic group of order n has exactly n elements: {e, g, g^2, ..., g^{n-1}}. 3. The structure of cyclic groups is central to understanding more complex group structures.
Applications of Group Theory
Group theory has applications across various fields such as: 1. Algebra: Provides insights into the structure of algebraic systems. 2. Physics: Used in the study of symmetry in physical systems and quantum mechanics. 3. Cryptography: Group theory underpins many cryptographic protocols, ensuring secure communication.
Lagrange’s theorem
Lagrange's theorem
Introduction to Lagrange's Theorem
Lagrange's theorem is a fundamental result in group theory, a branch of abstract algebra. It states that the order of any subgroup of a finite group divides the order of the group itself.
Statement of Lagrange's Theorem
If G is a finite group and H is a subgroup of G, then the order of H (denoted |H|) divides the order of G (denoted |G|). It can be expressed as |G| = |H| * [G:H], where [G:H] is the index of H in G.
Consequences of Lagrange's Theorem
The theorem has several important consequences, including the fact that the order of any element in a group must divide the order of the group.
Examples of Lagrange's Theorem
1. Symmetric group S3: The order of S3 is 6. The subgroups are of orders 1, 2, and 3, confirming Lagrange's theorem. 2. Cyclic groups: In a cyclic group of order n, all subgroups have orders that are divisors of n.
Applications of Lagrange's Theorem
Lagrange's theorem is used in proving other theorems in group theory, including the classification of finite abelian groups and the isomorphism theorems.
Conclusion
Lagrange's theorem provides a critical insight into the structure of finite groups and remains a cornerstone of abstract algebra.
Normal subgroups and quotient groups
Normal subgroups and quotient groups
Normal Subgroups
A normal subgroup is a subgroup N of a group G such that for all elements g in G, the conjugate gNg^{-1} is contained in N. This property is crucial for forming quotient groups. Normal subgroups allow for the definition of group actions and facilitate the study of homomorphisms.
Properties of Normal Subgroups
1. The trivial group and the whole group are always normal subgroups. 2. The intersection of normal subgroups is also a normal subgroup. 3. If N and M are normal subgroups of G, then their product NM is also a normal subgroup.
Quotient Groups
Given a group G and a normal subgroup N, the quotient group G/N consists of the cosets of N in G. The operation on the quotient group is defined by (gN)(hN) = (gh)N. Quotient groups play a vital role in group theory, as they help to understand the structure of groups through the lens of simpler groups.
Properties of Quotient Groups
1. If G is a group and N is a normal subgroup of G, then G/N is a group. 2. The order of the quotient group G/N is equal to the order of G divided by the order of N, |G/N| = |G| / |N|. 3. The first isomorphism theorem states that if φ: G -> H is a homomorphism, then G/ker(φ) is isomorphic to im(φ).
Ring theory and ideals
Ring theory and ideals
Introduction to Ring Theory
Ring theory is the study of algebraic structures known as rings. A ring consists of a set equipped with two binary operations that generalize the arithmetic operations of addition and multiplication.
Definition of a Ring
A ring is defined as a set R along with two operations: addition (+) and multiplication (·) such that: 1. (R, +) is an abelian group, 2. (R, ·) is a semigroup, 3. Distributive law holds: a · (b + c) = a · b + a · c and (a + b) · c = a · c + b · c for all a, b, c in R.
Types of Rings
There are several types of rings, such as: 1. Commutative Rings: Rings where multiplication is commutative. 2. Rings with Unity: Rings that have a multiplicative identity element. 3. Division Rings: Rings where every non-zero element has a multiplicative inverse. 4. Fields: Commutative rings with unity in which every non-zero element is invertible.
Ideals in Ring Theory
An ideal is a special subset of a ring that allows us to perform a kind of division. An ideal I of a ring R is a subset such that: 1. I is a subgroup of (R, +). 2. For every r in R and every a in I, the product r · a and a · r are also in I.
Types of Ideals
Ideals can be classified into two main types: 1. Left Ideal: A subset I of R is a left ideal if for every r in R and a in I, the product r · a is in I. 2. Right Ideal: A subset I of R is a right ideal if for every r in R and a in I, the product a · r is in I.
Maximal and Prime Ideals
1. Maximal Ideal: An ideal I of a ring R is maximal if I ≠ R and there are no other ideals between I and R. 2. Prime Ideal: An ideal P of a ring R is prime if whenever a · b is in P, then either a is in P or b is in P.
Factor Rings
Given a ring R and an ideal I, the factor ring R/I is the set of equivalence classes of R under the equivalence relation defined by I. This structure maintains the operations defined in R.
Applications of Ring Theory and Ideals
Ring theory has applications in various branches of mathematics, including number theory, algebraic geometry, and functional analysis. Ideals play a crucial role in constructing quotient structures and in studying the properties of rings.
Euclidean rings
Euclidean rings
Definition of Euclidean Rings
A Euclidean ring is a type of integral domain that allows a form of division with remainder. Specifically, for any two elements a and b in the ring, where b is not zero, there exist elements q and r in the ring such that a = bq + r with either r = 0 or a specific condition on the size of r.
Properties of Euclidean Rings
Euclidean rings possess several important properties including: 1. They are integral domains. 2. Every Euclidean ring is a principal ideal domain, as every ideal can be generated by a single element. 3. They facilitate the development of algorithms for finding greatest common divisors.
Examples of Euclidean Rings
Common examples of Euclidean rings include: 1. The ring of integers Z. 2. The ring of polynomials over a field, R[x]. These examples showcase how division with remainder can be performed.
Applications of Euclidean Rings
Euclidean rings play a crucial role in number theory and algebra. They are used in algorithms for computing greatest common divisors (GCD), and in the development of modular arithmetic techniques.
Comparison with Other Rings
Euclidean rings differ from more general types of rings in that not all rings maintain a division algorithm. Comparison can be made with rings such as principal ideal domains and unique factorization domains, which share some, but not all, properties with Euclidean rings.
