1.2 Subgroups, cosets, and normal subgroups
2 min read•july 25, 2024
Group structure and subgroups form the backbone of group theory. We'll explore how subgroups fit within larger groups, their properties, and how to identify them. This knowledge is crucial for understanding more complex group theory concepts.
We'll also dive into cosets, both left and right, and how they partition groups. Normal subgroups, a special type of , play a key role in advanced group theory and are essential for constructing quotient groups.
Group Structure and Subgroups
Subgroups and their examples
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- Subgroup definition embodies subset H of group G maintaining group properties under G's operation
- Criteria for subgroup requires closure under operation, identity element inclusion, inverse element existence
- Even integers under addition form subgroup of integers preserving additive structure
- Rotations constitute subgroup of symmetry group for regular polygon (hexagon)
- Special Linear Group SL(n,R) embeds as subgroup in General Linear Group GL(n,R) preserving determinant 1
- Trivial subgroups encompass identity subgroup {e} and entire group G itself always present
Subgroup criterion for proof
- Subgroup criterion (one-step test) verifies in H for all a, b in H
- Proving subgroup involves assuming arbitrary a, b in subset, demonstrating in subset, concluding criterion satisfaction
- Disproving subgroup requires finding specific a, b in subset, showing not in subset, concluding criterion failure
- Alternative methods include two-step test (closure, inverse existence) and three-step test (closure, associativity, inverse existence)
Left and right cosets
- aH comprises elements {ah : h ∈ H} for subgroup H of G and a in G
- Ha consists of elements {ha : h ∈ H} for subgroup H of G and a in G
- Cosets partition group into equal-sized subsets of order |H|
- states subgroup order divides group order:
- Coset representatives serve as stand-ins for entire coset (any element can represent)
Normal vs non-normal subgroups
- Normal subgroup N of G satisfies gN = Ng for all g in G, ensuring left and right coset equality
- Normal subgroups remain invariant under conjugation: for all g in G
- Center of group, kernel of , and subgroups of index 2 exemplify normal subgroups
- Non-normal subgroups include rotation subgroup in non-cyclic symmetry group and certain stabilizer subgroups in permutation groups
Conditions for normal subgroups
- Equivalent normality conditions: gH = Hg, , subgroup as kernel of homomorphism
- Proving normality requires showing gH ⊆ Hg and Hg ⊆ gH, or and for all g in G
- Normal subgroups enable quotient group construction and underpin homomorphism and theorem study
Key Terms to Review (15)
Abelian group: An abelian group is a set equipped with an operation that satisfies four fundamental properties: closure, associativity, identity, and inverses, while also ensuring that the operation is commutative. This means that for any two elements in the group, the order in which they are combined does not affect the result. The commutative property is crucial as it differentiates abelian groups from general groups, impacting how we understand subgroups, cosets, and normal subgroups.
Character of a representation: The character of a representation is a function that assigns to each group element the trace of the corresponding linear transformation in a representation of that group. It is an important tool in understanding how representations behave, especially when analyzing subgroups and the relationships between different representations through concepts like induction and restriction.
Cyclic Subgroup: A cyclic subgroup is a subset of a group that can be generated by a single element, meaning every element in the subgroup can be expressed as some integer power of that generator. This concept connects to the broader ideas of subgroups, cosets, and normal subgroups as it highlights how groups can be decomposed into smaller, manageable parts that still retain the group's structure.
First Isomorphism Theorem: The first isomorphism theorem states that if there is a homomorphism between two groups, then the image of the homomorphism is isomorphic to the quotient of the domain by the kernel of the homomorphism. This theorem establishes a vital connection between homomorphic mappings and the structure of groups, revealing how a homomorphism can relate groups through their substructures. The theorem provides a foundational framework for understanding how groups can be analyzed and compared through their homomorphic relationships.
G:h: In group theory, the notation $$g:h$$ denotes the set of left cosets of a subgroup $$H$$ in a group $$G$$, particularly focusing on the elements related by the action of the group. This concept is essential for understanding how subgroups partition a group into equal-sized parts and allows for a structured way to analyze group behavior and properties.
H ≤ g: In group theory, the notation 'h ≤ g' indicates that subgroup 'h' is a subgroup of group 'g', meaning that every element of 'h' is also an element of 'g'. This relationship establishes a foundational concept in understanding the structure and classification of groups, as well as how they relate to one another through subgroups. Recognizing this relationship is crucial for exploring other concepts like cosets and normal subgroups, which further illustrate how groups can be partitioned and how their internal symmetries operate.
Homomorphism: A homomorphism is a structure-preserving map between two algebraic structures, such as groups, rings, or vector spaces, that respects the operations defined on those structures. This concept is crucial for understanding how different algebraic entities relate to each other, especially when exploring their properties and behaviors under transformations.
Isomorphism: Isomorphism refers to a structural-preserving mapping between two algebraic structures, such as groups, that allows for the preservation of operations and relationships. This concept is vital in understanding how different mathematical systems can be equivalent in structure, enabling the classification of groups and representations based on their essential properties.
Lagrange's Theorem: Lagrange's Theorem states that in a finite group, the order of a subgroup divides the order of the group. This fundamental result highlights the relationship between a group and its subgroups, providing essential insights into the structure and properties of groups. It connects to concepts such as group actions, cosets, and normal subgroups, establishing a framework for understanding how groups can be decomposed and analyzed through their subgroup structures.
Left Coset: A left coset of a subgroup in a group is the set formed by multiplying each element of the subgroup by a fixed element from the group on the left. This concept is crucial in understanding how groups can be partitioned and helps illustrate the structure of groups and their subgroups, particularly when discussing normal subgroups and factor groups.
Representation of a Group: A representation of a group is a way to express group elements as linear transformations or matrices acting on a vector space. This concept allows us to study groups by analyzing their actions on different spaces, making it easier to understand their structure and properties. It connects deeply with the ideas of subgroups and normal subgroups, as well as the way group actions can be seen through cosets and their relationships in larger groups.
Right Coset: A right coset of a subgroup in a group is formed by taking an element from the group and multiplying it by every element in the subgroup. This concept helps to understand how groups can be partitioned into smaller, more manageable pieces and reveals important properties related to the structure of the group itself. Right cosets are particularly useful when discussing factor groups and their relationships to normal subgroups.
Simple Group: A simple group is a nontrivial group that does not have any normal subgroups other than the trivial group and itself. This definition highlights the fundamental nature of simple groups in the study of group theory, as they serve as the building blocks for more complex groups through group extensions. Simple groups are crucial in understanding the structure of finite groups and play a significant role in representation theory, especially in classifying and analyzing larger groups.
Subgroup: A subgroup is a subset of a group that itself forms a group under the same operation. This means that a subgroup must contain the identity element, be closed under the group operation, and contain the inverse of each of its elements. Understanding subgroups is essential as they help to analyze the structure of groups, reveal symmetries, and facilitate the study of representations and their properties.
Symmetric group: The symmetric group, denoted as $$S_n$$, is the group of all permutations of a finite set of $$n$$ elements, capturing the essence of rearranging objects. This group is fundamental in understanding how groups act on sets, with its elements representing all possible ways to rearrange the members of the set, leading to various applications in algebra and combinatorics.