libterm.com A subgroup of a group is a subset that itself forms a group under the original operation, while a coset is formed by multiplying all elements of a subgroup by a fixed element from the group. Cosets partition the group into equal-sized, non-overlapping sets, which are fundamental in understanding group structure and quotient groups. Explore the rest of the article to deepen your understanding of how subgroups and cosets interplay within algebraic systems.
Table of Comparison
| Aspect | Subgroup | Coset |
|---|---|---|
| Definition | A subset of a group that is itself a group under the same operation. | A translation of a subgroup by a fixed group element; form a partition of the group. |
| Structure | Closed under group operation and inverses; contains the identity element. | Not necessarily a subgroup; may lack identity and closure. |
| Notation | H <= G (H is a subgroup of G) | gH or Hg, for g in G and subgroup H |
| Cardinality | Size divides the size of the group (Lagrange's Theorem). | Same cardinality as the subgroup it translates. |
| Example | H = {e, a, a2} <= G | gH = {g, ga, ga2} |
| Role in Group Theory | Fundamental for analyzing group structure. | Used to study coset partitions, quotient groups, and indices. |
Introduction to Subgroups and Cosets
Subgroups are subsets of a group that themselves form a group under the same operation, maintaining closure, associativity, identity, and invertibility. Cosets arise by translating a subgroup within the group, forming sets like left cosets \( gH = \{gh \mid h \in H\} \) or right cosets \( Hg = \{hg \mid h \in H\} \), where \( g \) is an element of the group and \( H \) is the subgroup. The partition of a group into distinct cosets relative to a subgroup reveals important structural properties, including index and normality.
Definition of Subgroup
A subgroup is a subset of a group that itself forms a group under the same operation, satisfying closure, identity, inverses, and associativity. A coset, however, is formed by taking an element of the group and combining it with every element of a subgroup, resulting in either a left or right coset. The defining property of a subgroup distinguishes it from cosets, as subgroups maintain group structure internally, whereas cosets represent partitions of the group based on subgroup membership.
Definition of Coset
A coset is defined as the set formed by multiplying all elements of a subgroup by a fixed element from the parent group, resulting in either a left coset or a right coset depending on the multiplication order. Unlike a subgroup, a coset does not necessarily satisfy the subgroup criteria such as closure and the presence of the identity element. Cosets play a crucial role in partitioning groups into equal-sized, non-overlapping subsets that facilitate the study of group structure and the application of Lagrange's theorem.
Key Differences Between Subgroup and Coset
A subgroup is a subset of a group that itself forms a group under the same operation, meaning it contains the identity element, is closed under the operation, and includes inverses for all its elements. A coset, formed by multiplying a fixed element of the group with every element of a subgroup, is not necessarily a subgroup as it may not contain the identity element or inverses. Key differences lie in their structural properties: subgroups satisfy all group axioms, while cosets are translations of subgroups that partition the group without inherently forming groups themselves.
Properties of Subgroups
Subgroups are subsets of groups that satisfy closure, associativity, identity, and inverse properties inherent to the parent group, ensuring they form a group themselves. A coset is formed by multiplying a fixed element with every element of a subgroup, partitioning the group into equal-sized, non-overlapping subsets. Subgroups are characterized by their ability to retain group structure within the larger group, while cosets capture the idea of equivalence classes under the subgroup's action.
Properties of Cosets
Cosets of a subgroup H in a group G partition G into disjoint subsets, each having the same cardinality as H, reflecting the index of H in G. Unlike subgroups, cosets need not contain the identity element and generally lack closure under group operation, making them non-subgroups unless coinciding with H itself. Left and right cosets may differ in non-abelian groups, illustrating the distinction between normal and non-normal subgroups through their coset structures.
Examples of Subgroups
A subgroup is a smaller group contained within a larger group that itself satisfies the group properties, such as the set of even integers within the group of all integers under addition. A coset is formed by multiplying all elements of a subgroup by a fixed element from the larger group, like the set of all integers plus 1 forming a coset of the even integers subgroup. Examples of subgroups include the set of rotations in a symmetry group and the subgroup of multiples of 3 in the integers under addition.
Examples of Cosets
Cosets arise from a subgroup \( H \) of a group \( G \), formed by multiplying each element of \( H \) by a fixed element \( g \in G \). For example, if \( G = \mathbb{Z} \) (integers under addition) and \( H = 3\mathbb{Z} \) (multiples of 3), the cosets are \( 0 + H = 3\mathbb{Z} \), \( 1 + H = \{ \dots, -2, 1, 4, 7, \dots \} \), and \( 2 + H = \{ \dots, -1, 2, 5, 8, \dots \} \). These cosets partition the group \( G \) into disjoint subsets, highlighting the structure of \( G \) relative to the subgroup \( H \).
Applications in Group Theory
Subgroups serve as foundational elements within group theory, defining subsets that themselves satisfy group properties, which are crucial for analyzing group structure. Cosets, formed by multiplying a fixed element with all elements of a subgroup, facilitate the study of group partitions and lead to important results such as Lagrange's theorem. Applications in group theory leverage cosets to understand normality, quotient groups, and symmetry classifications in algebraic systems.
Summary: Coset vs Subgroup
A subgroup is a subset of a group that itself satisfies the group axioms, meaning it contains the identity element, is closed under the group operation, and includes inverses. A coset, formed by multiplying all elements of a subgroup by a fixed group element, partitions the group into equal-sized, disjoint subsets but does not necessarily form a subgroup. Understanding the distinction between subgroups and cosets is essential for grasping group structure, with subgroups maintaining internal group properties and cosets providing a way to analyze group actions and quotient groups.
Subgroup - Coset Infographic
