Lección 2 - Orden e índice de un subgrupo | Estructuras Algebraicas | UNED

Understanding Group Order and Element Order

Definition of Group Order

• The order of a group refers to the number of elements within that group, which is crucial for understanding its structure.

• Care must be taken as the term "order" also applies to individual elements, leading to potential confusion in notation.

• The cardinality of a group can be denoted using various notations such as |G| or #G, indicating the total count of elements.

Definition of Element Order

• The order of an element in a group is defined as the smallest natural number n such that raising the element to n yields the identity element.

• This definition emphasizes that for multiplicative groups, this means finding n where g^n = 1 , with 1 representing the identity.

Properties Related to Element Order

• The identity element has an order of 1 since it only needs to be multiplied by itself once to yield itself.

• There exists a unique element in any group with order 1, which is always the identity element.

Introduction to Torsion Elements

Definition and Characteristics

• An element is termed a torsion element if it generates a finite subgroup; this indicates that its order is finite.

• Conversely, if an element has infinite order, it does not generate a finite subgroup.

Properties of Torsion Elements

• If an element a in G is torsion, then there exists some natural number n , such that a^n = e , where e is the identity.

• The inverse of any torsion element also possesses finite order and shares its order with the original element.

Index in Subgroups

Defining Index

• The index of a subgroup H subseteq G , denoted as [G:H], represents how many distinct cosets exist when dividing group G by subgroup H.

• It can be understood as comparing cardinalities: |G| divided by |H| gives insight into their relative sizes.

Equivalence Relations and Classes

• To define indices properly, equivalence relations are established based on membership in subgroups.

• These equivalence classes partition the entire group into subsets where each subset corresponds uniquely to cosets formed by multiplying elements from H.

Lagrange's Theorem

Fundamental Insights

• Lagrange's theorem states that for any finite group G and its subgroup H, both orders (the size/count of elements in each set respectively), divide one another.

• Specifically, if both orders are finite, then |G| = |H| * [G:H], establishing clear relationships between them.

This structured approach provides clarity on key concepts related to groups and their properties while ensuring easy navigation through timestamps for further exploration.

Introduction to Infinite Order and Lagrange's Theorem

Working with Subgroups

• Discusses the concept of infinite order in relation to Lagrange's theorem, focusing on subgroups contained within a group G .

• Emphasizes that the orders of two subgroups H and K must be coprime (i.e., their greatest common divisor is 1), leading to an intersection containing only the identity element.

• Explains that for a subgroup intersection H cap K , if it contains only the identity, then both subgroups are disjoint except for this element.

Transitiveness in Indices

• Introduces transitiveness properties related to indices of subgroups, which will be useful later in understanding finite field structures.

• States that if subgroup H is contained within subgroup K , then both indices are finite when one is finite.

Index Relationships Among Subgroups

Finite Indices

• Describes how the index of subgroup H in group G can be expressed through indices involving other subgroups like K .

• Presents a formula relating cardinalities of groups and their respective indices, emphasizing its straightforward nature.

Constructing New Subgroups

Product of Subgroups

• Discusses constructing new subgroups from existing ones by taking their product, highlighting conditions under which this holds true.

• Analyzes the cardinality of the product set formed by two finite order subgroups and how it relates to their intersection.

Cyclic Groups

Definition and Properties

• Defines cyclic groups as those generated by a single element, noting their significance in group theory.

• Highlights that groups of interest are those generated by a finite set of elements leading back to cyclic structures.

Characterizing Cyclic Groups

Conditions for Cyclicity

• Establishes criteria for determining whether a group is cyclic based on relationships between orders of elements.

• Clarifies that for a group to be cyclic, there must exist an element whose order matches the group's overall order.

Abelian Property in Cyclic Groups

Commutativity

• Notes that cyclic groups inherently possess commutative properties, meaning operations within them do not depend on order.

Existence of Subgroups

Implications from Lagrange’s Theorem

• Discusses implications regarding finding subgroups based on divisibility conditions derived from Lagrange’s theorem.

All Subgroup Properties

• States every subgroup within a cyclic group retains its own cyclic property. This means any chosen subgroup will also be cyclic regardless of its size or structure.

Minimal Generating Sets

• Introduces minimal generating sets as essential components for understanding infinitely generated groups.

Finding Minimal Generators

• Discusses challenges associated with identifying minimal generating systems within finitely generated groups.

Bounding Minimal Generators

• Concludes with results bounding the maximum number of elements required for minimal generating systems based on group size.