# 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):
$d\in \mathrm{derived_subgroup}\left(G\right)\equiv \exists x,y,n.\mathrm{length}\left(x\right)=n\wedge \mathrm{length}\left(y\right)=n\wedge \forall i.\mathrm{entry}\left(x,i\right)\in \mathrm{carrier}\left(G\right)\wedge \mathrm{entry}\left(y,i\right)\in \mathrm{carrier}\left(G\right)\wedge \mathrm{expression}\left(G,\mathrm{apply_to_list}\left(×,\mathrm{list}\left(\mathrm{entry}\left(x,i\right)\mathrm{entry}\left(y,i\right){\mathrm{entry}\left(x,i\right)}^{-1}{\mathrm{entry}\left(y,i\right)}^{-1}|i\in \left[1,n\right]\right)\right)\right)=d$
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):
$d\in \mathrm{center}\left(G\right)\equiv \forall g.g\in \mathrm{carrier}\left(G\right)⇒\left(\mathrm{multiplication}\left(G\right)\right)\left(d,g\right)=\left(\mathrm{multiplication}\left(G\right)\right)\left(g,d\right)$
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):
$d\in \mathrm{centralizer}\left(G,g\right)\equiv \left(d\in \mathrm{carrier}\left(G\right)\wedge \left(\mathrm{multiplication}\left(G\right)\right)\left(d,g\right)=\left(\mathrm{multiplication}\left(G\right)\right)\left(g,d\right)\right)$
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:
$\mathrm{free_group}\left(\left(a,b\right)\right)$
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):
$d\in \mathrm{normalizer}\left(G,X\right)\equiv \left(d\in \mathrm{carrier}\left(G\right)\wedge \left(\mathrm{multiplication}\left(G\right)\right)\left(d,X\right)=\left(\mathrm{multiplication}\left(G\right)\right)\left(X,d\right)\right)$
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):
$\mathrm{carrier}\left(\mathrm{symmetric_groupn}\left(n\right)\right)=\mathrm{permutationsn}\left(n\right)$
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

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

## 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

 [Next: invertibles] [Previous: alternating_group] [Top]

## 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

 [First: automorphism_group] [Previous: alternatingn] [Top]