OpenMath Content Dictionary: group4

Canonical URL:

http://www.openmath.org/cd/group4.ocd

CD File:

group4.ocd

CD as XML Encoded OpenMath:

group4.omcd

Defines:

are_conjugate, conjugacy_class, conjugacy_class_representatives, conjugacy_classes, left_coset, left_coset_representative, left_cosets, left_transversal, right_coset, right_coset_representative, right_cosets, right_transversal

Date:

2004-06-01

Version:

1 (Revision 2)

Review Date:

2006-06-01

Status:

experimental

A CD of sets constructed from groups

Written by Arjeh M. Cohen 2004-03-02.
Edited AMC 2004-03-05

conjugacy_classes

Description:

This symbol represents a unary function whose argument should be a group. Its value on a group is the set of conjugacy classes of that group.

Signatures:

sts

conjugacy_class_representatives

Description:

This symbol represents a unary function whose argument should be a group. Its value on a group is a set of representatives of the conjugacy classes of that group.

Signatures:

sts

are_conjugate

Description:

This symbol represents a boolean ternary function whose first argument is a group G and whose second and third arguments are elements x and y of G. Its value on G, x, and y is true if and only if x and y are conjugate in G.

Commented Mathematical property (CMP):

x and y are conjugate if and only if there is h in G such that x = h y h^(-1).

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA><OMS cd="logic1" name="equivalent"/>
       <OMA><OMS cd="group4" name="are_conjugate"/>
            <OMV name="G"/>
            <OMV name="x"/> <OMV name="y"/>
       </OMA>
       <OMBIND><OMS cd="quant1" name="exists"/>
            <OMBVAR><OMV name="h"/></OMBVAR>
            <OMA><OMS cd="logic1" name="and"/>
                 <OMA><OMS cd="set1" name="in"/>
                      <OMV name="h"/>
                      <OMA><OMS cd="group1" name="carrier"/>
                           <OMV name="G"/>
                      </OMA>
                 </OMA>
                 <OMA><OMS cd="relation1" name="eq"/>
                      <OMV name="x"/>
                      <OMA><OMS cd="group1" name="expression"/>
                           <OMV name="G"/>
                           <OMA><OMS cd="arith1" name="times"/>
                                <OMV name="h"/>
                                <OMV name="y"/>
                                <OMA><OMS cd="arith1" name="power"/>
                                     <OMV name="h"/>
                                     <OMI>-1</OMI>
                                </OMA>
                           </OMA>
                      </OMA>
                 </OMA>
            </OMA>
        </OMBIND>
   </OMA>
</OMOBJ>

equivalent (are_conjugate ( G, x, y) , exists [ h] . (and (in ( h, carrier ( G) ) , eq ( x, expression ( G, times ( h, y, power ( h, -1) ) ) ) ) ) )

$$\mathrm{are\_conjugate}(G,x,y)\equiv \exists \hspace{0.17em}h.h\in \mathrm{carrier}\left(G\right)\wedge x=\mathrm{expression}(G,hy{h}^{-1})$$

Signatures:

sts

conjugacy_class

Description:

This symbol represents a binary function, whose first argument is a group G and whose second argument is an element x of G. Its value on G and x is the set of elements which are conjugate to x in G.

Commented Mathematical property (CMP):

The conjugacy class in G of h is the subset {g^(-1) h g | g in G} of G.

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA><OMS cd="relation1" name="eq"/>
       <OMA><OMS cd="group4" name="conjugacy_class"/>
            <OMV name="G"/>   <OMV name="h"/>
       </OMA>
       <OMA><OMS cd="set1" name="suchthat"/>
            <OMA><OMS cd="group1" name="carrier"/>
                 <OMV name="G"/>
            </OMA>
            <OMBIND><OMS cd="fns1" name="lambda"/>
                 <OMBVAR> <OMV name="x"/></OMBVAR>
                 <OMA><OMS cd="group4" name="are_conjugate"/>
                      <OMV name="G"/><OMV name="x"/><OMV name="h"/>
                 </OMA>
            </OMBIND>
       </OMA>
  </OMA>
</OMOBJ>

eq (conjugacy_class ( G, h) , suchthat (carrier ( G) , lambda [ x] . (are_conjugate ( G, x, h) ) ) )

$$\mathrm{conjugacy\_class}(G,h)=\left\{x\in \mathrm{carrier}\left(G\right)\right|\mathrm{are\_conjugate}(G,x,h)\}$$

Signatures:

sts

left_transversal

Description:

The binary function whose value is a set of representatives for the left cosets of the second argument as a subgroup of the first.

Signatures:

sts

right_transversal

Description:

The binary function whose value is a set of representatives for the right cosets of the second argument as a subgroup of the first.

Signatures:

sts

left_coset

Description:

This symbol represents a ternary function whose first argument is a group G, whose second argument is a subgroup H of G, and whose third argument is an element x of G. Its value on G, H, and x is the left coset of H in G containing x, that is, the set x H.

Signatures:

sts

right_coset

Description:

This symbol represents a ternary function whose first argument is a group G, whose second argument is a subgroup H of G, and whose third argument is an element x of G. Its value on G, H, and x is the right coset of H in G containing x, that is, the set H x.

Signatures:

sts

left_cosets

Description:

The binary function whose value is the set of left cosets of the second argument in the first.

Signatures:

sts

right_cosets

Description:

The binary function whose value is the set of right cosets of the second argument in the first.

Signatures:

sts

left_coset_representative

Description:

This symbol represents a quaternary function whose first argument is a group G, whose second argument is a subgroup H of G, whose third argument is left_transversal T of H in G, and whose fourth argument is an element of G. It assigns to G, H, T, g the element of t of T representing the left coset of H containing g, that is, t H = g H .

Signatures:

sts

right_coset_representative

Description:

This symbol represents a quaternary function whose first argument is a group G, whose second argument is a subgroup H of G, whose third argument is right_transversal T of H in G, and whose fourth argument is an element of G. It assigns to G, H, T, g the element of t of T representing the right coset of H containing g, that is, H t = H g.

Signatures:

sts