Subgroup

Left cosets of Z 2 in Z 8

Symmetric group 4; Cayley table; numbers

Alternating group 4; Cayley table; numbers

Klein four-group; Cayley table; subgroup of S4 (elements 0,7,16,23)

Cyclic group 3; Cayley table; subgroup of S4 (elements 0,3,4)

Cyclic group 3; Cayley table; subgroup of S4 (elements 0,11,19)

Subgroup in the context of mathematics, specifically within the branch of abstract algebra, refers to a group that is contained within another group, known as the parent group or simply the larger group. The concept of subgroups is fundamental in the study of group theory, as it allows mathematicians to understand the structure of groups by analyzing their smaller, constituent parts.

Definition

A subgroup H of a group G is a subset of G that is itself a group under the operation defined on G. Formally, for H to be a subgroup of G, it must satisfy the following criteria:

• Closure: If a and b are elements of H, then the product ab is also in H.
• Identity: The identity element e of G is also in H.
• Inverses: For every element a in H, there exists an inverse a⁻¹ in H such that aa⁻¹ = e, where e is the identity element of G.

Types of Subgroups

Subgroups can be classified into various types based on their properties and the way they relate to the parent group:

• Normal subgroups: A subgroup H is normal in G if it is invariant under conjugation by elements of G. This means for every a in H and every g in G, the element gag⁻¹ is also in H. Normal subgroups are important because they allow the construction of quotient groups.
• Cyclic subgroups: These are subgroups generated by a single element. A cyclic subgroup of G is denoted by ⟨a⟩, where a is an element of G. Every element of a cyclic subgroup can be written as aⁿ for some integer n.
• Sylow subgroups: In the context of finite groups, Sylow subgroups are maximal subgroups of a particular order. They are central to the Sylow theorems, which provide conditions under which groups contain subgroups of given orders.

Examples

• In the group of integers under addition, Z, any set of integers that can be written as nZ (where n is a non-zero integer) forms a subgroup. For example, the even integers form a subgroup of Z.
• In the group of permutations of a set (the symmetric group), the set of permutations that leave at least one element fixed forms a subgroup.

Subgroup Lattice

The collection of all subgroups of a group G, including G itself and the trivial group (containing only the identity element), can be organized into a structure known as a subgroup lattice. This lattice visually represents the inclusion relationships between subgroups of G.

Applications

Subgroups play a crucial role in many areas of mathematics and its applications. They are used in the study of symmetry in chemistry and physics, the analysis of polynomial roots in algebra, and in the classification of finite groups, among other areas.