Groups, subgroups, and actions form the foundation of abstract algebra. These concepts help us understand the structure and symmetry in mathematical systems, from simple number operations to complex geometric transformations.
By studying these ideas, we gain powerful tools for analyzing and solving problems in various fields. Group theory's applications range from cryptography to quantum mechanics, making it a crucial area of study in modern mathematics and science.
Groups and their properties
Definition and axioms
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A group is a set G together with a binary operation ∗ satisfying the group axioms:
Closure: For all a,b∈G, a∗b∈G
Associativity: For all a,b,c∈G, (a∗b)∗c=a∗(b∗c)
Identity: There exists an element e∈G such that a∗e=e∗a=a for all a∈G
Inverses: For each a∈G, there exists an element a−1∈G such that a∗a−1=a−1∗a=e
Group properties
The order of a group is the number of elements in the group
A group is finite if its order is finite, and infinite otherwise
A group is abelian (or commutative) if a∗b=b∗a for all a,b∈G
Examples of abelian groups include (Z,+), (R,+), and (R∗,⋅)
A group is cyclic if it can be generated by a single element
For each positive integer n, there exists a unique of order n (up to isomorphism), denoted Z/nZ or Zn
The center of a group G is the set of elements that commute with every element of G
The center is always a of G
The direct product of two groups (G,∗) and (H,∘) is the group (G×H,⋅) where the operation ⋅ is defined by (g1,h1)⋅(g2,h2)=(g1∗g2,h1∘h2)
Subgroups and their relationships
Subgroup definition and test
A subset H of a group G is a subgroup if H is itself a group under the same operation as G
The subgroup test states that H is a subgroup of G if and only if H is nonempty and closed under the group operation and inverses
The trivial subgroup {e} and the group G itself are always subgroups of G
A proper subgroup is a subgroup that is not the whole group
Cosets and Lagrange's Theorem
If H is a subgroup of G, then the left cosets of H in G are the sets of the form aH={ah:h∈H} for each a∈G
Similarly, the right cosets are the sets Ha={ha:h∈H}
In general, left and right cosets may differ, but they coincide when H is a
states that if G is a finite group and H is a subgroup of G, then the order of H divides the order of G
As a consequence, the order of any element of a finite group divides the order of the group
The index of a subgroup H in G, denoted [G:H], is the number of left (or right) cosets of H in G
Lagrange's Theorem implies that [G:H]=∣G∣/∣H∣
Normal subgroups
A normal subgroup is a subgroup N of G such that gNg−1=N for all g∈G
Every subgroup of an is normal
The kernel of a group homomorphism is always a normal subgroup of the domain
Group actions
Definition and properties
A of a group G on a set X is a function ⋅:G×X→X satisfying the following properties:
e⋅x=x for all x∈X, where e is the identity element of G
(gh)⋅x=g⋅(h⋅x) for all g,h∈G and x∈X
The of an element x∈X under the action of G is the set Orb(x)={g⋅x:g∈G}
The orbits form a partition of X
The of an element x∈X is the set Stab(x)={g∈G:g⋅x=x}
The stabilizer is always a subgroup of G
Orbit-Stabilizer Theorem and types of actions
The Orbit-Stabilizer Theorem states that for any x∈X, there is a bijection between the orbit of x and the set of left cosets of Stab(x) in G
As a consequence, ∣Orb(x)∣=[G:Stab(x)]
A group action is faithful if the only group element that fixes every x∈X is the identity
Equivalently, the action is faithful if the homomorphism from G to Sym(X) defined by the action is injective
A group action is transitive if for any x,y∈X, there exists g∈G such that g⋅x=y
Equivalently, the action is transitive if it has only one orbit
Group theorems
Subgroup and normal subgroup theorems
Prove that the intersection of two subgroups is also a subgroup
Prove that the center of a group is a normal subgroup
Prove that the kernel of a group homomorphism is a normal subgroup of the domain
Prove that if G is a group and N is a normal subgroup of G, then the quotient group G/N, consisting of cosets of N in G, is a well-defined group under the operation (aN)(bN)=(ab)N
Isomorphism theorems and Cayley's Theorem
Prove the First Isomorphism Theorem: If ϕ:G→H is a group homomorphism, then G/ker(ϕ)≅im(ϕ)
Prove Cayley's Theorem: Every group G is isomorphic to a subgroup of the symmetric group acting on G
Prove that if a group G acts transitively on a set X and H is the stabilizer of a point x∈X, then there is a bijection between X and the set of left cosets of H in G
Key Terms to Review (13)
Abelian group: An abelian group is a set equipped with an operation that combines any two elements to form a third element, satisfying four fundamental properties: closure, associativity, identity, and invertibility. The defining characteristic of an abelian group is that the operation is commutative, meaning the order in which you combine elements does not affect the outcome. This property links abelian groups to various concepts in mathematics, particularly in the study of symmetry and structure within algebraic systems.
Cyclic Group: A cyclic group is a group that can be generated by a single element, where every element of the group can be expressed as some power (or multiple) of this generator. This concept is fundamental in understanding the structure of groups, as cyclic groups serve as building blocks for more complex groups and play a key role in various mathematical areas, including number theory and abstract algebra.
Évariste Galois: Évariste Galois was a French mathematician known for his groundbreaking work in abstract algebra and the foundations of Galois Theory, which connects field theory and group theory. His contributions laid the groundwork for understanding the solvability of polynomial equations, highlighting the relationship between field extensions and symmetry.
Group: A group is a set equipped with a binary operation that satisfies four fundamental properties: closure, associativity, the existence of an identity element, and the existence of inverses for each element. Groups provide a structure for understanding symmetry, operations, and algebraic systems, making them crucial in various mathematical contexts such as algebra, geometry, and number theory.
Group Action: A group action is a formal way in which a group systematically interacts with a set, where each element of the group corresponds to a specific transformation of that set. This concept allows for the exploration of symmetry and structure within mathematical objects, showing how group elements can rearrange or map elements in a way that preserves some underlying structure. Understanding group actions is crucial as it connects to various concepts like orbits and stabilizers, and it plays a significant role in the study of normal subgroups and quotient groups.
Lagrange's Theorem: Lagrange's Theorem states that for any finite group, the order of a subgroup divides the order of the group. This theorem is fundamental in understanding the structure of groups and their subgroups, as it provides insight into how these smaller sets relate to the whole. The theorem emphasizes that the number of elements in a subgroup must be a factor of the number of elements in the group, revealing crucial properties about both sets and aiding in the classification of groups.
Niels Henrik Abel: Niels Henrik Abel was a Norwegian mathematician known for his groundbreaking contributions to various areas of mathematics, particularly in the field of algebra. His work laid foundational principles that influenced Galois Theory and helped to shape our understanding of polynomial equations and their solvability.
Normal Subgroup: A normal subgroup is a subgroup that remains invariant under conjugation by elements of the larger group, meaning that for any element in the normal subgroup and any element in the group, the conjugate will also lie in that subgroup. This property ensures that normal subgroups play a crucial role in the construction of quotient groups, where the larger group can be 'factored' by the normal subgroup. Their significance extends into Galois theory, where normality relates to field extensions and symmetries.
Orbit: In the context of group actions, an orbit is the set of elements that a particular element can be transformed into under the action of a group. This concept connects the behavior of a group on a set and illustrates how different elements are related through group actions, revealing the structure and symmetries of the set.
Stabilizer: A stabilizer is a subgroup of a group that keeps certain elements unchanged during a group action. It identifies the elements of the group that act as symmetries for a given object or set, helping to understand the structure and properties of the group through its action on various sets. The concept is essential in analyzing how groups interact with other mathematical structures, leading to deeper insights about both the groups and the objects they act upon.
Subgroup: A subgroup is a subset of a group that itself satisfies the group properties, meaning it is closed under the group operation and contains the identity element as well as the inverses of its elements. Subgroups play a critical role in understanding the structure of groups, including how they can interact with each other through operations like normality and forming quotient groups.
Sylow Theorems: The Sylow Theorems are a set of important results in group theory that provide detailed information about the structure of finite groups, specifically regarding the existence, conjugacy, and number of Sylow p-subgroups. These theorems link the properties of a finite group to its order and prime factors, helping us understand how subgroups can be formed and behave within larger groups.
Transitive Action: Transitive action refers to a group action on a set where, if one element can be transformed into another by the action of a group element, then any element can be transformed into any other through a series of such actions. This concept highlights the ability of the group to move elements around within the set, emphasizing that the entire set can be reached from any single point through the group's operations. It connects to several important concepts including orbit, stabilizer, and homomorphism, showcasing how groups interact with sets and their structure.