OpenMath Content Dictionary: groupname1

Canonical URL:

http://www.openmath.org/cd/groupname1.ocd

CD Base:

http://www.openmath.org/cd

CD File:

groupname1.ocd

CD as XML Encoded OpenMath:

groupname1.omcd

Defines:

dihedral_group, quaternion_group, cyclic_group, generalized_quaternion_group

Date:

2004-06-01

Version:

1 (Revision 2)

Review Date:

2006-06-01

Status:

experimental

Well known groups in group theory

Written by Arjeh M. Cohen 2003-04-15

quaternion_group

Description:

This symbol represents the quaternion group of order 8.

Commented Mathematical property (CMP):

The quaternion group is isomorphic to the group generated by a, b with presentation a^2 = b^2 = aba^(-1)b^(-1) and a^4 = 1.

Formal Mathematical property (FMP):

OpenMath XML (source)

  <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
    <OMA><OMS cd="group2" name="isomorphic"/>
              <OMS cd="groupname1" name="quaternion_group"/>
              <OMA><OMS cd="group3" name="quotient_group"/>
                   <OMA><OMS cd="group3" name="free_group"/>
                        <OMV name="a"/>  <OMV name="b"/>
                   </OMA>
                   <OMA><OMS cd="group1" name="normal_closure"/>
                        <OMA><OMS cd="group3" name="free_group"/>
                             <OMV name="a"/>  <OMV name="b"/>
                        </OMA>
                        <OMA><OMS cd="fns2" name="apply_to_list"/>
                             <OMBIND><OMS cd="fns1" name="lambda"/>
                                 <OMBVAR><OMV name="x"/> </OMBVAR>
                                 <OMA><OMS cd="group1" name="expression"/>
                                      <OMA><OMS cd="group3" name="free_group"/>
                                           <OMV name="a"/>  <OMV name="b"/>
                                      </OMA>
                                      <OMV name="x"/>
                                 </OMA>
                             </OMBIND>
                             <OMA><OMS cd="list1" name="list"/>
                                  <OMA><OMS cd="arith1" name="power"/>
                                       <OMV name="a"/>
                                       <OMI>4</OMI>
                                  </OMA>
                                  <OMA><OMS cd="arith1" name="times"/>
                                       <OMA><OMS cd="arith1" name="power"/>
                                            <OMV name="a"/>
                                            <OMI>2</OMI>
                                       </OMA>
                                       <OMA><OMS cd="arith1" name="power"/>
                                            <OMV name="b"/>
                                            <OMI>-2</OMI>
                                       </OMA>
                                  </OMA>

                                  <OMA><OMS cd="arith1" name="times"/>
                                       <OMA><OMS cd="arith1" name="power"/>
                                            <OMV name="a"/>
                                            <OMI>2</OMI>
                                       </OMA>
                                       <OMV name="b"/>
                                       <OMV name="a"/>
                                       <OMA><OMS cd="arith1" name="power"/>
                                            <OMV name="b"/>
                                            <OMI>-1</OMI>
                                       </OMA>
                                       <OMA><OMS cd="arith1" name="power"/>
                                            <OMV name="a"/>
                                            <OMI>-1</OMI>
                                       </OMA>
                                  </OMA>
                             </OMA>
                        </OMA>
                   </OMA>
              </OMA>
    </OMA>
  </OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="group2">isomorphic</csymbol>
  <csymbol cd="groupname1">quaternion_group</csymbol>
  <apply><csymbol cd="group3">quotient_group</csymbol>
   <apply><csymbol cd="group3">free_group</csymbol><ci>a</ci><ci>b</ci></apply>
   <apply><csymbol cd="group1">normal_closure</csymbol>
    <apply><csymbol cd="group3">free_group</csymbol><ci>a</ci><ci>b</ci></apply>
    <apply><csymbol cd="fns2">apply_to_list</csymbol>
     <bind><csymbol cd="fns1">lambda</csymbol>
      <bvar><ci>x</ci></bvar>
      <apply><csymbol cd="group1">expression</csymbol>
       <apply><csymbol cd="group3">free_group</csymbol><ci>a</ci><ci>b</ci></apply>
       <ci>x</ci>
      </apply>
     </bind>
     <apply><csymbol cd="list1">list</csymbol>
      <apply><csymbol cd="arith1">power</csymbol><ci>a</ci><cn>4</cn></apply>
      <apply><csymbol cd="arith1">times</csymbol>
       <apply><csymbol cd="arith1">power</csymbol><ci>a</ci><cn>2</cn></apply>
       <apply><csymbol cd="arith1">power</csymbol><ci>b</ci><cn>-2</cn></apply>
      </apply>
      <apply><csymbol cd="arith1">times</csymbol>
       <apply><csymbol cd="arith1">power</csymbol><ci>a</ci><cn>2</cn></apply>
       <ci>b</ci>
       <ci>a</ci>
       <apply><csymbol cd="arith1">power</csymbol><ci>b</ci><cn>-1</cn></apply>
       <apply><csymbol cd="arith1">power</csymbol><ci>a</ci><cn>-1</cn></apply>
      </apply>
     </apply>
    </apply>
   </apply>
  </apply>
 </apply>
</math>

Prefix

isomorphic (quaternion_group, quotient_group (free_group ( a, b) , normal_closure (free_group ( a, b) , apply_to_list (lambda [ x ] . (expression (free_group ( a, b) , x) ) , list (power ( a, 4) , times (power ( a, 2) , power ( b, -2) ) , times (power ( a, 2) , b, a, power ( b, -1) , power ( a, -1) ) ) ) ) ) )

Popcorn

group2.isomorphic(groupname1.quaternion_group, group3.quotient_group(group3.free_group($a, $b), group1.normal_closure(group3.free_group($a, $b), fns2.apply_to_list(fns1.lambda[$x -> group1.expression(group3.free_group($a, $b), $x)], [$a ^ 4 , $a ^ 2 * $b ^ -2 , $a ^ 2 * $b * $a * $b ^ -1 * $a ^ -1]))))

Rendered Presentation MathML

$$\mathrm{isomorphic}(\mathrm{quaternion\_group},\mathrm{quotient\_group}(\mathrm{free\_group}(a,b),\mathrm{normal\_closure}(\mathrm{free\_group}(a,b),\mathrm{apply\_to\_list}(\lambda \hspace{0.17em}x.\mathrm{expression}(\mathrm{free\_group}(a,b),x),\begin{array}{c}{a}^4\hfill \\ a^2{b}^{-2}\hfill \\ a^{2}ba{b}^{-1}{a}^{-1}\hfill \end{array}))))$$

Commented Mathematical property (CMP):

The center of Q has order 2.

Commented Mathematical property (CMP):

The derived subgroup of Q coincides with the center of Q.

Signatures:

sts

dihedral_group

Description:

This symbol is a function with one argument, which should be a positive integer n. When applied to n it represents the dihedral group of order 2n. This is the group of all isometries (including reflections) of the regular n-gon in the plane.

Commented Mathematical property (CMP):

The dihedral group of order 2n is isomorphic to the group generated by a, b with presentation a^2 = b^n = 1 and a b a = b^(-1).

Formal Mathematical property (FMP):

OpenMath XML (source)

  <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
    <OMA><OMS cd="group2" name="isomorphic"/>
              <OMA><OMS cd="groupname1" name="dihedral_group"/>
                   <OMV name="n"/>
              </OMA>
              <OMA><OMS cd="group3" name="quotient_group"/>
                   <OMA><OMS cd="group3" name="free_group"/>
                        <OMV name="a"/>  <OMV name="b"/>
                   </OMA>
                   <OMA><OMS cd="group1" name="normal_closure"/>
                        <OMA><OMS cd="group3" name="free_group"/>
                             <OMV name="a"/>  <OMV name="b"/>
                        </OMA>
                        <OMA><OMS cd="fns2" name="apply_to_list"/>
                             <OMBIND><OMS cd="fns1" name="lambda"/>
                                 <OMBVAR><OMV name="x"/> </OMBVAR>
                                 <OMA><OMS cd="group1" name="expression"/>
                                      <OMA><OMS cd="group3" name="free_group"/>
                                           <OMV name="a"/>  <OMV name="b"/>
                                      </OMA>
                                      <OMV name="x"/>
                                 </OMA>
                             </OMBIND>
                             <OMA><OMS cd="list1" name="list"/>
                                  <OMA><OMS cd="arith1" name="power"/>
                                       <OMV name="a"/>
                                       <OMI>2</OMI>
                                  </OMA>
                                  <OMA><OMS cd="arith1" name="power"/>
                                       <OMV name="b"/>
                                       <OMV name="n"/>
                                  </OMA>
                                  <OMA><OMS cd="arith1" name="times"/>
                                       <OMV name="a"/>
                                       <OMV name="b"/>
                                       <OMV name="a"/>
                                       <OMV name="b"/>
                                  </OMA>
                             </OMA>
                        </OMA>
                   </OMA>
              </OMA>
    </OMA>
  </OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="group2">isomorphic</csymbol>
  <apply><csymbol cd="groupname1">dihedral_group</csymbol><ci>n</ci></apply>
  <apply><csymbol cd="group3">quotient_group</csymbol>
   <apply><csymbol cd="group3">free_group</csymbol><ci>a</ci><ci>b</ci></apply>
   <apply><csymbol cd="group1">normal_closure</csymbol>
    <apply><csymbol cd="group3">free_group</csymbol><ci>a</ci><ci>b</ci></apply>
    <apply><csymbol cd="fns2">apply_to_list</csymbol>
     <bind><csymbol cd="fns1">lambda</csymbol>
      <bvar><ci>x</ci></bvar>
      <apply><csymbol cd="group1">expression</csymbol>
       <apply><csymbol cd="group3">free_group</csymbol><ci>a</ci><ci>b</ci></apply>
       <ci>x</ci>
      </apply>
     </bind>
     <apply><csymbol cd="list1">list</csymbol>
      <apply><csymbol cd="arith1">power</csymbol><ci>a</ci><cn>2</cn></apply>
      <apply><csymbol cd="arith1">power</csymbol><ci>b</ci><ci>n</ci></apply>
      <apply><csymbol cd="arith1">times</csymbol>
       <ci>a</ci>
       <ci>b</ci>
       <ci>a</ci>
       <ci>b</ci>
      </apply>
     </apply>
    </apply>
   </apply>
  </apply>
 </apply>
</math>

Prefix

isomorphic (dihedral_group ( n) , quotient_group (free_group ( a, b) , normal_closure (free_group ( a, b) , apply_to_list (lambda [ x ] . (expression (free_group ( a, b) , x) ) , list (power ( a, 2) , power ( b, n) , times ( a, b, a, b) ) ) ) ) )

Popcorn

group2.isomorphic(groupname1.dihedral_group($n), group3.quotient_group(group3.free_group($a, $b), group1.normal_closure(group3.free_group($a, $b), fns2.apply_to_list(fns1.lambda[$x -> group1.expression(group3.free_group($a, $b), $x)], [$a ^ 2 , $b ^ $n , $a * $b * $a * $b]))))

Rendered Presentation MathML

$$\mathrm{isomorphic}(\mathrm{dihedral\_group}\left(n\right),\mathrm{quotient\_group}(\mathrm{free\_group}(a,b),\mathrm{normal\_closure}(\mathrm{free\_group}(a,b),\mathrm{apply\_to\_list}(\lambda \hspace{0.17em}x.\mathrm{expression}(\mathrm{free\_group}(a,b),x),(a^2,b^n,abab)))))$$

Signatures:

sts

cyclic_group

Description:

This symbol is a function with one argument, which should be a natural number n. When applied to n it represents the cyclic group of order n.

Signatures:

sts

generalized_quaternion_group

Description:

This symbol is a function with one argument, which should be a positive integer. When applied to n it represents the generalized quaternion group of order 4n. This is the group with three generators a, b, and c and relations c = a^2 = b^n, c*a = a*c , b*c = c*b, a*b = b*a*c, and c^2 = 1.

Signatures:

sts