OpenMath Content Dictionary: group3

Canonical URL:

http://www.openmath.org/cd/group3.ocd

CD Base:

http://www.openmath.org/cd

CD File:

group3.ocd

CD as XML Encoded OpenMath:

group3.omcd

Defines:

GL, GLn, SL, SLn, alternating_group, alternatingn, automorphism_group, center, centralizer, derived_subgroup, direct_power, direct_product, free_group, invertibles, normalizer, quotient_group, sylow_subgroup, symmetric_group, symmetric_groupn

Date:

2004-06-01

Version:

1 (Revision 2)

Review Date:

2006-06-01

Status:

experimental

---

A CD of group constructions

Written by Arjeh M. Cohen 2004-02-20.

---

automorphism_group

Description:

This is a function with a single argument which must be a group. It refers to the automorphism group of its argument.

Signatures:

sts

---

[Next: direct_product] [Last: invertibles] [Top]

---

direct_product

Description:

This is an n-ary function whose arguments must be groups. It refers to the direct product of its arguments.

Signatures:

sts

---

[Next: direct_power] [Previous: automorphism_group] [Top]

---

direct_power

Description:

This is a binary function whose first argument should be a group G and whose second argument should be a natural number n. It refers to the direct product of n copies of G.

Signatures:

sts

---

[Next: sylow_subgroup] [Previous: direct_product] [Top]

---

sylow_subgroup

Description:

This symbol represents a binary function with two arguments, the first is a group G and the second a prime number p. When applied to G and p, it represents a Sylow p-subgroup of G (which is unique up to conjugacy in G).

Signatures:

sts

---

[Next: derived_subgroup] [Previous: direct_power] [Top]

---

derived_subgroup

Description:

The unary function whose value is the subgroup of argument generated by all products of the form xyx^-1y^-1.

Commented Mathematical property (CMP):

d in the derived subgroup of G if and only if there exist lists x,y of elements of G of equal length such that d is the product x_1 y_1 x_1^(-1) y_1^(-1) ... x_n y_n x_n^(-1) y_n^(-1).

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA><OMS cd="logic1" name="equivalent"/>
       <OMA><OMS cd="set1" name="in"/>
            <OMV name="d"/>
            <OMA><OMS cd="group3" name="derived_subgroup"/>
                 <OMV name="G"/>
            </OMA>
       </OMA>
       <OMBIND><OMS cd="quant1" name="exists"/>
            <OMBVAR><OMV name="x"/><OMV name="y"/><OMV name="n"/></OMBVAR>
            <OMA><OMS cd="logic1" name="and"/>
                 <OMA><OMS cd="relation1" name="eq"/>
                      <OMA><OMS cd="list3" name="length"/>
                           <OMV name="x"/>
                      </OMA>
                      <OMV name="n"/>
                 </OMA>
                 <OMA><OMS cd="relation1" name="eq"/>
                      <OMA><OMS cd="list3" name="length"/>
                           <OMV name="y"/>
                      </OMA>
                      <OMV name="n"/>
                 </OMA>
                 <OMBIND><OMS cd="quant1" name="forall"/>
                      <OMBVAR><OMV name="i"/> </OMBVAR>
                      <OMA><OMS cd="logic1" name="and"/>
                           <OMA><OMS cd="set1" name="in"/>
                                <OMA><OMS cd="list3" name="entry"/>
                                     <OMV name="x"/> <OMV name="i"/>
                                </OMA>
                                <OMA><OMS cd="group1" name="carrier"/>
                                     <OMV name="G"/>
                                </OMA>
                           </OMA>
                           <OMA><OMS cd="set1" name="in"/>
                                <OMA><OMS cd="list3" name="entry"/>
                                     <OMV name="y"/> <OMV name="i"/>
                                </OMA>
                                <OMA><OMS cd="group1" name="carrier"/>
                                     <OMV name="G"/>
                                </OMA>
                           </OMA>
                      </OMA>
                 </OMBIND>
                 <OMA><OMS cd="relation1" name="eq"/>
                      <OMA><OMS cd="group1" name="expression"/>
                           <OMV name="G"/>
                           <OMA><OMS cd="fns2" name="apply_to_list"/>
                                <OMS cd="arith1" name="times"/>
                                <OMA><OMS cd="list1" name="map"/>
                                     <OMBIND><OMS cd="fns1" name="lambda"/>
                                         <OMBVAR><OMV name="i"/> </OMBVAR>
                                         <OMA><OMS cd="arith1" name="times"/>
                                              <OMA><OMS cd="list3" name="entry"/>
                                                   <OMV name="x"/> <OMV name="i"/>
                                              </OMA>
                                              <OMA><OMS cd="list3" name="entry"/>
                                                   <OMV name="y"/> <OMV name="i"/>
                                              </OMA>
   
                                              <OMA><OMS cd="arith1" name="power"/>
                                                   <OMA><OMS cd="list3" name="entry"/>
                                                        <OMV name="x"/> <OMV name="i"/>
                                                   </OMA>
                                                   <OMI>-1</OMI>
                                              </OMA>
                                              <OMA><OMS cd="arith1" name="power"/>
                                                   <OMA><OMS cd="list3" name="entry"/>
                                                        <OMV name="y"/> <OMV name="i"/>
                                                   </OMA>
                                                   <OMI>-1</OMI>
                                              </OMA>
                                         </OMA>
                                    </OMBIND>
                                    <OMA><OMS cd="interval1" name="integer_interval"/>
                                         <OMI>1</OMI> 
                                         <OMV name="n"/>
                                    </OMA>
                                 </OMA>
                            </OMA>
                       </OMA>
                       <OMV name="d"/>
                  </OMA>
             </OMA>
        </OMBIND>
   </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="logic1">equivalent</csymbol>
  <apply><csymbol cd="set1">in</csymbol>
   <ci>d</ci>
   <apply><csymbol cd="group3">derived_subgroup</csymbol><ci>G</ci></apply>
  </apply>
  <bind><csymbol cd="quant1">exists</csymbol>
   <bvar><ci>x</ci></bvar>
   <bvar><ci>y</ci></bvar>
   <bvar><ci>n</ci></bvar>
   <apply><csymbol cd="logic1">and</csymbol>
    <apply><csymbol cd="relation1">eq</csymbol>
     <apply><csymbol cd="list3">length</csymbol><ci>x</ci></apply>
     <ci>n</ci>
    </apply>
    <apply><csymbol cd="relation1">eq</csymbol>
     <apply><csymbol cd="list3">length</csymbol><ci>y</ci></apply>
     <ci>n</ci>
    </apply>
    <bind><csymbol cd="quant1">forall</csymbol>
     <bvar><ci>i</ci></bvar>
     <apply><csymbol cd="logic1">and</csymbol>
      <apply><csymbol cd="set1">in</csymbol>
       <apply><csymbol cd="list3">entry</csymbol><ci>x</ci><ci>i</ci></apply>
       <apply><csymbol cd="group1">carrier</csymbol><ci>G</ci></apply>
      </apply>
      <apply><csymbol cd="set1">in</csymbol>
       <apply><csymbol cd="list3">entry</csymbol><ci>y</ci><ci>i</ci></apply>
       <apply><csymbol cd="group1">carrier</csymbol><ci>G</ci></apply>
      </apply>
     </apply>
    </bind>
    <apply><csymbol cd="relation1">eq</csymbol>
     <apply><csymbol cd="group1">expression</csymbol>
      <ci>G</ci>
      <apply><csymbol cd="fns2">apply_to_list</csymbol>
       <csymbol cd="arith1">times</csymbol>
       <apply><csymbol cd="list1">map</csymbol>
        <bind><csymbol cd="fns1">lambda</csymbol>
         <bvar><ci>i</ci></bvar>
         <apply><csymbol cd="arith1">times</csymbol>
          <apply><csymbol cd="list3">entry</csymbol><ci>x</ci><ci>i</ci></apply>
          <apply><csymbol cd="list3">entry</csymbol><ci>y</ci><ci>i</ci></apply>
          <apply><csymbol cd="arith1">power</csymbol>
           <apply><csymbol cd="list3">entry</csymbol><ci>x</ci><ci>i</ci></apply>
           <cn type="integer">-1</cn>
          </apply>
          <apply><csymbol cd="arith1">power</csymbol>
           <apply><csymbol cd="list3">entry</csymbol><ci>y</ci><ci>i</ci></apply>
           <cn type="integer">-1</cn>
          </apply>
         </apply>
        </bind>
        <apply><csymbol cd="interval1">integer_interval</csymbol><cn type="integer">1</cn><ci>n</ci></apply>
       </apply>
      </apply>
     </apply>
     <ci>d</ci>
    </apply>
   </apply>
  </bind>
 </apply>
</math>

Prefix

equivalent (in ( d, derived_subgroup ( G) ) , exists [ x y n] . (and (eq (length ( x) , n) , eq (length ( y) , n) , forall [ i ] . (and (in (entry ( x, i) , carrier ( G) ) , in (entry ( y, i) , carrier ( G) ) ) ) , eq (expression ( G, apply_to_list (times, map (lambda [ i ] . (times (entry ( x, i) , entry ( y, i) , power (entry ( x, i) , -1) , power (entry ( y, i) , -1) ) ) , integer_interval (1, n) ) ) ) , d) ) ) )

Popcorn

logic1.equivalent(set1.in($d, group3.derived_subgroup($G)), quant1.exists[$x, $y, $n -> list3.length($x) = $n and list3.length($y) = $n and quant1.forall[$i -> set1.in(list3.entry($x, $i), group1.carrier($G)) and set1.in(list3.entry($y, $i), group1.carrier($G))] and group1.expression($G, fns2.apply_to_list(arith1.times, list1.map(fns1.lambda[$i -> list3.entry($x, $i) * list3.entry($y, $i) * list3.entry($x, $i) ^ -1 * list3.entry($y, $i) ^ -1], interval1.integer_interval(1, $n)))) = $d])

Rendered Presentation MathML

$$d\in \mathrm{derived\_subgroup}\left(G\right)\equiv \exists \hspace{0.17em}x,y,n.\mathrm{length}\left(x\right)=n\wedge \mathrm{length}\left(y\right)=n\wedge \forall \hspace{0.17em}i.\mathrm{entry}(x,i)\in \mathrm{carrier}\left(G\right)\wedge \mathrm{entry}(y,i)\in \mathrm{carrier}\left(G\right)\wedge \mathrm{expression}(G,\mathrm{apply\_to\_list}(\times ,\mathrm{list}\left(\mathrm{entry}(x,i)\mathrm{entry}(y,i){\mathrm{entry}(x,i)}^{-1}{\mathrm{entry}(y,i)}^{-1}\right|i\in [1,n])))=d$$

Signatures:

sts

---

[Next: quotient_group] [Previous: sylow_subgroup] [Top]

---

quotient_group

Description:

The binary function whose value is the factor group of the first argument by the second, assuming the second is normal in the first.

Signatures:

sts

---

[Next: center] [Previous: derived_subgroup] [Top]

---

center

Description:

This symbols represents a unary function whose argument should be a group G. Its value is the biggest subgroup of G all of whose elements commute with all elements of G.

Commented Mathematical property (CMP):

d is in the center of G if and only if for all g in G we have g d= d g.

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA><OMS cd="logic1" name="equivalent"/>
       <OMA><OMS cd="set1" name="in"/>
            <OMV name="d"/>
            <OMA><OMS cd="group3" name="center"/>
                 <OMV name="G"/>
            </OMA>
       </OMA>
       <OMBIND><OMS cd="quant1" name="forall"/>
            <OMBVAR><OMV name="g"/></OMBVAR>
            <OMA><OMS cd="logic1" name="implies"/>
                 <OMA><OMS cd="set1" name="in"/>
                      <OMV name="g"/>
                      <OMA><OMS cd="group1" name="carrier"/>
                           <OMV name="G"/>
                      </OMA>
                 </OMA> 
                 <OMA><OMS cd="relation1" name="eq"/>
                      <OMA><OMA><OMS cd="group1" name="multiplication"/>
                                <OMV name="G"/>
                           </OMA>
                           <OMV name="d"/>    <OMV name="g"/>
                      </OMA>
                      <OMA><OMA><OMS cd="group1" name="multiplication"/>
                                <OMV name="G"/>
                           </OMA>
                           <OMV name="g"/> <OMV name="d"/>
                      </OMA>
                 </OMA>
            </OMA>
       </OMBIND>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="logic1">equivalent</csymbol>
  <apply><csymbol cd="set1">in</csymbol>
   <ci>d</ci>
   <apply><csymbol cd="group3">center</csymbol><ci>G</ci></apply>
  </apply>
  <bind><csymbol cd="quant1">forall</csymbol>
   <bvar><ci>g</ci></bvar>
   <apply><csymbol cd="logic1">implies</csymbol>
    <apply><csymbol cd="set1">in</csymbol>
     <ci>g</ci>
     <apply><csymbol cd="group1">carrier</csymbol><ci>G</ci></apply>
    </apply>
    <apply><csymbol cd="relation1">eq</csymbol>
     <apply>
      <apply><csymbol cd="group1">multiplication</csymbol><ci>G</ci></apply>
      <ci>d</ci>
      <ci>g</ci>
     </apply>
     <apply>
      <apply><csymbol cd="group1">multiplication</csymbol><ci>G</ci></apply>
      <ci>g</ci>
      <ci>d</ci>
     </apply>
    </apply>
   </apply>
  </bind>
 </apply>
</math>

Prefix

equivalent (in ( d, center ( G) ) , forall [ g] . (implies (in ( g, carrier ( G) ) , eq (multiplication ( G) ( d, g) , multiplication ( G) ( g, d) ) ) ) )

Popcorn

logic1.equivalent(set1.in($d, group3.center($G)), quant1.forall[$g -> set1.in($g, group1.carrier($G)) ==> group1.multiplication($G)($d, $g) = group1.multiplication($G)($g, $d)])

Rendered Presentation MathML

$$d\in \mathrm{center}\left(G\right)\equiv \forall \hspace{0.17em}g.g\in \mathrm{carrier}\left(G\right)\Rightarrow \left(\mathrm{multiplication}\left(G\right)\right)(d,g)=\left(\mathrm{multiplication}\left(G\right)\right)(g,d)$$

Signatures:

sts

---

[Next: centralizer] [Previous: quotient_group] [Top]

---

centralizer

Description:

This symbols represents a binary function whose first argument should be a group G and whose second argument should be an element g or a list of elements L of the group G. Its value is the subgroup of G of all elements commuting with g or, if the second argument is a list, all elements of L.

Commented Mathematical property (CMP):

d is in the centralizer of g in G if and only if g d= d g.

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA><OMS cd="logic1" name="equivalent"/>
       <OMA><OMS cd="set1" name="in"/>
            <OMV name="d"/>
            <OMA><OMS cd="group3" name="centralizer"/>
                 <OMV name="G"/>                 <OMV name="g"/>
            </OMA>
       </OMA>
       <OMA><OMS cd="logic1" name="and"/>
            <OMA><OMS cd="set1" name="in"/>
                 <OMV name="d"/>
                 <OMA><OMS cd="group1" name="carrier"/>
                      <OMV name="G"/>
                 </OMA>
            </OMA> 
            <OMA><OMS cd="relation1" name="eq"/>
                 <OMA><OMA><OMS cd="group1" name="multiplication"/>
                           <OMV name="G"/>
                      </OMA>
                      <OMV name="d"/>    <OMV name="g"/>
                 </OMA>
                 <OMA><OMA><OMS cd="group1" name="multiplication"/>
                           <OMV name="G"/>
                      </OMA>
                      <OMV name="g"/> <OMV name="d"/>
                 </OMA>
            </OMA>
       </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="logic1">equivalent</csymbol>
  <apply><csymbol cd="set1">in</csymbol>
   <ci>d</ci>
   <apply><csymbol cd="group3">centralizer</csymbol><ci>G</ci><ci>g</ci></apply>
  </apply>
  <apply><csymbol cd="logic1">and</csymbol>
   <apply><csymbol cd="set1">in</csymbol>
    <ci>d</ci>
    <apply><csymbol cd="group1">carrier</csymbol><ci>G</ci></apply>
   </apply>
   <apply><csymbol cd="relation1">eq</csymbol>
    <apply>
     <apply><csymbol cd="group1">multiplication</csymbol><ci>G</ci></apply>
     <ci>d</ci>
     <ci>g</ci>
    </apply>
    <apply>
     <apply><csymbol cd="group1">multiplication</csymbol><ci>G</ci></apply>
     <ci>g</ci>
     <ci>d</ci>
    </apply>
   </apply>
  </apply>
 </apply>
</math>

Prefix

equivalent (in ( d, centralizer ( G, g) ) , and (in ( d, carrier ( G) ) , eq (multiplication ( G) ( d, g) , multiplication ( G) ( g, d) ) ) )

Popcorn

logic1.equivalent(set1.in($d, group3.centralizer($G, $g)), set1.in($d, group1.carrier($G)) and group1.multiplication($G)($d, $g) = group1.multiplication($G)($g, $d))

Rendered Presentation MathML

$$d\in \mathrm{centralizer}(G,g)\equiv (d\in \mathrm{carrier}\left(G\right)\wedge \left(\mathrm{multiplication}\left(G\right)\right)(d,g)=\left(\mathrm{multiplication}\left(G\right)\right)(g,d))$$

Signatures:

sts

---

[Next: free_group] [Previous: center] [Top]

---

free_group

Description:

This symbol represents a unary function. The argument is a list or a set. When evaluated on such an argument, the function represents the free group generated by the entries of the list or set.

Example:

The free group on the letters a, b:

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA><OMS cd="group3" name="free_group"/>
       <OMA><OMS cd="list1" name="list"/>
            <OMV name="a"/>  <OMV name="b"/>
       </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="group3">free_group</csymbol>
  <apply><csymbol cd="list1">list</csymbol><ci>a</ci><ci>b</ci></apply>
 </apply>
</math>

Prefix

free_group (list ( a, b) )

Popcorn

group3.free_group([$a , $b])

Rendered Presentation MathML

$$\mathrm{free\_group}\left((a,b)\right)$$

Signatures:

sts

---

[Next: GL] [Previous: centralizer] [Top]

---

GL

Description:

This symbol is a function with one argument, which should be a vector space or a module V. When applied to V it represents the group of all invertible linear transformations of V.

Signatures:

sts

---

[Next: SL] [Previous: free_group] [Top]

---

SL

Description:

This symbol is a function with one argument, which should be a a module V over a commutative ring. When applied to V it represents the group of all invertible linear transformations of V of determinant 1.

Signatures:

sts

---

[Next: GLn] [Previous: GL] [Top]

---

GLn

Description:

This symbol is a function with two arguments. The first should be a positive integer n, the second a field F. When applied to n and F it represents the group of all invertible linear transformations of the vector space over F of dimension n.

Signatures:

sts

---

[Next: SLn] [Previous: SL] [Top]

---

SLn

Description:

This symbol is a function with two arguments. The first should be a positive integer n, the second a field F. When applied to n and F it represents the group of all invertible linear transformations of the vector space over F of dimension n having determinant 1.

Signatures:

sts

---

[Next: normalizer] [Previous: GLn] [Top]

---

normalizer

Description:

This symbols represents a binary function whose first argument should be a group G and whose second argument should be a set of elements or a subgroup L of the group G. Its value is the subgroup of G of all elements normalizing L.

Commented Mathematical property (CMP):

d is in the normalizer of X in G if and only if g X= X g.

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA><OMS cd="logic1" name="equivalent"/>
       <OMA><OMS cd="set1" name="in"/>
            <OMV name="d"/>
            <OMA><OMS cd="group3" name="normalizer"/>
                 <OMV name="G"/>       <OMV name="X"/>
            </OMA>
       </OMA>
       <OMA><OMS cd="logic1" name="and"/>
            <OMA><OMS cd="set1" name="in"/>
                 <OMV name="d"/>
                 <OMA><OMS cd="group1" name="carrier"/>
                      <OMV name="G"/>
                 </OMA>
            </OMA> 
            <OMA><OMS cd="relation1" name="eq"/>
                 <OMA><OMA><OMS cd="group1" name="multiplication"/>
                           <OMV name="G"/>
                      </OMA>
                      <OMV name="d"/>    <OMV name="X"/>
                 </OMA>
                 <OMA><OMA><OMS cd="group1" name="multiplication"/>
                           <OMV name="G"/>
                      </OMA>
                      <OMV name="X"/> <OMV name="d"/>
                 </OMA>
            </OMA>
       </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="logic1">equivalent</csymbol>
  <apply><csymbol cd="set1">in</csymbol>
   <ci>d</ci>
   <apply><csymbol cd="group3">normalizer</csymbol><ci>G</ci><ci>X</ci></apply>
  </apply>
  <apply><csymbol cd="logic1">and</csymbol>
   <apply><csymbol cd="set1">in</csymbol>
    <ci>d</ci>
    <apply><csymbol cd="group1">carrier</csymbol><ci>G</ci></apply>
   </apply>
   <apply><csymbol cd="relation1">eq</csymbol>
    <apply>
     <apply><csymbol cd="group1">multiplication</csymbol><ci>G</ci></apply>
     <ci>d</ci>
     <ci>X</ci>
    </apply>
    <apply>
     <apply><csymbol cd="group1">multiplication</csymbol><ci>G</ci></apply>
     <ci>X</ci>
     <ci>d</ci>
    </apply>
   </apply>
  </apply>
 </apply>
</math>

Prefix

equivalent (in ( d, normalizer ( G, X) ) , and (in ( d, carrier ( G) ) , eq (multiplication ( G) ( d, X) , multiplication ( G) ( X, d) ) ) )

Popcorn

logic1.equivalent(set1.in($d, group3.normalizer($G, $X)), set1.in($d, group1.carrier($G)) and group1.multiplication($G)($d, $X) = group1.multiplication($G)($X, $d))

Rendered Presentation MathML

$$d\in \mathrm{normalizer}(G,X)\equiv (d\in \mathrm{carrier}\left(G\right)\wedge \left(\mathrm{multiplication}\left(G\right)\right)(d,X)=\left(\mathrm{multiplication}\left(G\right)\right)(X,d))$$

Signatures:

sts

---

[Next: symmetric_group] [Previous: SLn] [Top]

---

symmetric_group

Description:

This symbol is a function with one argument, which should be a set X. When applied to a set X it represents the group of all permutations on X .

Signatures:

sts

---

[Next: symmetric_groupn] [Previous: normalizer] [Top]

---

symmetric_groupn

Description:

This symbol is a function with one argument, which should be a natural number n. When applied to n it represents the group of all permutations on the set {1,2,... ,n}.

Commented Mathematical property (CMP):

The carrier set of symmetric_groupn(k) consists of all permutations with support in the integers {1,...,k}.

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA><OMS cd="relation1" name="eq"/>
       <OMA><OMS cd="group1" name="carrier"/>
            <OMA><OMS cd="group3" name="symmetric_groupn"/>
                 <OMV name="n"/>
            </OMA>
       </OMA>
       <OMA><OMS cd="permutation1" name="permutationsn"/>
            <OMV name="n"/>
       </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="group1">carrier</csymbol>
   <apply><csymbol cd="group3">symmetric_groupn</csymbol><ci>n</ci></apply>
  </apply>
  <apply><csymbol cd="permutation1">permutationsn</csymbol><ci>n</ci></apply>
 </apply>
</math>

Prefix

eq (carrier (symmetric_groupn ( n) ) , permutationsn ( n) )

Popcorn

group1.carrier(group3.symmetric_groupn($n)) = permutation1.permutationsn($n)

Rendered Presentation MathML

$$\mathrm{carrier}\left(\mathrm{symmetric\_groupn}\left(n\right)\right)=\mathrm{permutationsn}\left(n\right)$$

Signatures:

sts

---

[Next: alternating_group] [Previous: symmetric_group] [Top]

---

alternating_group

Description:

This symbol is a function with one argument, which should be a set X. When applied to a set X it represents the group of all even permutations on X .

Signatures:

sts

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[Next: alternatingn] [Previous: symmetric_groupn] [Top]

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alternatingn

Description:

This symbol is a function with one argument, which should be a natural number n. When applied to n it represents the group of all even permutations on the set {1,2, ...,n}.

Signatures:

sts

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invertibles

Description:

This symbol is a function with one argument, which should be a monoid M. When applied to M it represents the group of all invertible elements of M.

Signatures:

sts

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