# OpenMath Content Dictionary: group4

Canonical URL:
http://www.openmath.org/cd/group4.ocd
CD Base:
http://www.openmath.org/cd
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

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## 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

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## 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):
$\mathrm{are_conjugate}\left(G,x,y\right)\equiv \exists h.h\in \mathrm{carrier}\left(G\right)\wedge x=\mathrm{expression}\left(G,hy{h}^{-1}\right)$
Signatures:
sts

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## 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):
$\mathrm{conjugacy_class}\left(G,h\right)=\left\{x\in \mathrm{carrier}\left(G\right)|\mathrm{are_conjugate}\left(G,x,h\right)\right\}$
Signatures:
sts

 [Next: left_transversal] [Previous: are_conjugate] [Top]

## 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

 [Next: right_transversal] [Previous: conjugacy_class] [Top]

## 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

 [Next: left_coset] [Previous: left_transversal] [Top]

## 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

 [Next: right_coset] [Previous: right_transversal] [Top]

## 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

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## left_cosets

Description:

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

Signatures:
sts

 [Next: right_cosets] [Previous: right_coset] [Top]

## right_cosets

Description:

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

Signatures:
sts

 [Next: left_coset_representative] [Previous: left_cosets] [Top]

## 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

 [Next: right_coset_representative] [Previous: right_cosets] [Top]

## 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

 [First: conjugacy_classes] [Previous: left_coset_representative] [Top]