## G-sets

Abbreviation: **Gset**

### Definition

A ** G-set** is a structure $\mathbf{A}=\langle A,f_g (g\in G)\rangle$, where $\langle G, \cdot, ^{-1}, 1\rangle$ is a group, such that

$f_1$ is the identity map: $1x=x$ and

the group action associates: $(g\cdot h)x=g(hx)$

Remark:
$f_g(x)=gx$ is a unary operation called ** the group action by $g$**.

If follows from the associativity that $f_{g^{-1}}$ is the inverse function of $f_g$.

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It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.

##### Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be … . A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x ... y)=h(x) ... h(y)$

### Definition

An ** …** is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle
...\rangle$ such that

$...$ is …: $axiom$

$...$ is …: $axiom$

### Examples

Example 1:

### Basic results

### Properties

Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

### Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ \end{array}$ $\begin{array}{lr} f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$

### Subclasses

[[...]] subvariety

[[...]] expansion

### Superclasses

[[...]] supervariety

[[...]] subreduct

### References

^{1)}F. Lastname,

**, Journal,**

*Title***1**, 23–45 MRreview

Trace: » g-sets