vakioaliryhmä

Vakioaliryhmä refers to a concept in abstract algebra, specifically within group theory. It describes a special type of subgroup of a group. A subgroup H of a group G is called a vakioaliryhmä if it is invariant under conjugation by every element of G. This means that for any element g in G and any element h in H, the element g h g⁻¹ is also in H. This property is often expressed as gHg⁻¹ = H for all g ∈ G.

Vakioaliryhmät are also known as normal subgroups or invariant subgroups. The term "vakioaliryhmä" is a Finnish term that translates directly to "constant subgroup" or "invariant subgroup." The concept is fundamental to understanding the structure of groups and is crucial for defining quotient groups, also known as factor groups.

A key characteristic of vakioaliryhmät is that they form the basis for homomorphisms. If φ: G → K is a group homomorphism, then the kernel of φ, denoted as ker(φ), is always a vakioaliryhmä of G. Conversely, every vakioaliryhmä of G can be realized as the kernel of some homomorphism. The trivial subgroup (containing only the identity element) and the group G itself are always vakioaliryhmät of G. Groups whose only vakioaliryhmät are the trivial subgroup and the group itself are called simple groups.