OpenMath Content Dictionary: group5

Canonical URL:

http://www.openmath.org/cd/group5.ocd

CD File:

group5.ocd

CD as XML Encoded OpenMath:

group5.omcd

Defines:

homomorphism_by_generators, left_quotient_map, right_quotient_map

Date:

2004-07-07

Version:

1 (Revision 1)

Review Date:

2006-06-01

Status:

experimental

A CD of functions for relating group elements to their images in quotients.

Written by Arjeh M. Cohen 2004-07-07

right_quotient_map

Description:

This symbol is a binary function whose first argument is a group G and whose second argument is an subgroup H of G. When applied to G and H, its value is the natural quotient map from G to the quotient group G/H, sending x to the left coset xH of G.

Commented Mathematical property (CMP):

The image of an element x is the left coset of x with respect to H.

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA><OMS cd="relation1" name="eq"/>
       <OMA><OMA><OMS cd="group5" name="right_quotient_map"/>
                 <OMV name="G"/><OMV name="H"/>
            </OMA>
            <OMV name="x"/>
       </OMA>
       <OMA><OMS cd="group4" name="left_coset"/>
            <OMV name="G"/><OMV name="H"/>
            <OMV name="x"/>
       </OMA>
  </OMA>
</OMOBJ>

eq (right_quotient_map ( G, H) ( x) , left_coset ( G, H, x) )

$$\left(\mathrm{right\_quotient\_map}(G,H)\right)\left(x\right)=\mathrm{left\_coset}(G,H,x)$$

Signatures:

sts

left_quotient_map

Description:

This symbol is a binary function whose first argument is a group G and whose second argument is an subgroup H of G. When applied to G and H, its value is the natural quotient map from G to the quotient group G/H, sending x to the right coset Hx of G.

Commented Mathematical property (CMP):

The image of an element x is the right coset of x with respect to H.

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA><OMS cd="relation1" name="eq"/>
       <OMA><OMA><OMS cd="group5" name="left_quotient_map"/>
                 <OMV name="G"/><OMV name="H"/>
            </OMA>
            <OMV name="x"/>
       </OMA>
       <OMA><OMS cd="group4" name="right_coset"/>
            <OMV name="G"/><OMV name="H"/>
            <OMV name="x"/>
       </OMA>
  </OMA>
</OMOBJ>

eq (left_quotient_map ( G, H) ( x) , right_coset ( G, H, x) )

$$\left(\mathrm{left\_quotient\_map}(G,H)\right)\left(x\right)=\mathrm{right\_coset}(G,H,x)$$

Signatures:

sts

The left and right quotients have a natural group structure if and only if H
is a normal subgroup of G.

homomorphism_by_generators

Description:

This is a function with three arguments the first two of which must be groups F and K. The third argument should be a set or a list L of ordered pairs (lists of length 2). Each pair [x,y] from L consists of an element x from F and an element y from K. When applied to F, K, and L, the symbol represents the group homomorphism from F to K that maps the first entry x of each pair [x,y] to the second entry y of the same pair.

Signatures:

sts