Group with operators

Groups with operators were extensively studied by Emmy Noether and her school in the 1920s. She employed the concept in her original formulation of the three Noether isomorphism theorems.

Definition

A group with operators $(G, Ω)$ can be defined as a group $G = (G, \cdot)$ together with an action of a set $Ω$ on $G$ :

Ω \times G \to G : (ω, g) \mapsto g^{ω}

that is distributive relative to the group law:

(g \cdot h)^{ω} = g^{ω} \cdot h^{ω} .

For each $ω \in Ω$ , the map $g \mapsto g^{ω}$ is then an endomorphism of G. From this, it results that a Ω-group can also be viewed as a group G with an indexed family ${(u_{ω})}_{ω \in Ω}$ of endomorphisms of G.

$Ω$ is called the operator domain. The associate endomorphisms are called the homotheties of G.

Given two groups G, H with same operator domain $Ω$ , a homomorphism of groups with operators from $(G, Ω)$ to $(H, Ω)$ is a group homomorphism $ϕ : G \to H$ satisfying

ϕ (g^{ω}) = (ϕ (g))^{ω}

for all

ω \in Ω

and

g \in G .

A subgroup S of G is called a stable subgroup, $Ω$ -subgroup or $Ω$ -invariant subgroup if it respects the homotheties, that is

s^{ω} \in S

for all

s \in S

and

ω \in Ω .

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Definition

Category-theoretic remarks

Examples

Applications

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