# OpenMath Content Dictionary: group5

Canonical URL:
http://www.openmath.org/cd/group5.ocd
CD Base:
http://www.openmath.org/cd
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):
$\left(\mathrm{right_quotient_map}\left(G,H\right)\right)\left(x\right)=\mathrm{left_coset}\left(G,H,x\right)$
Signatures:
sts

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## 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):
$\left(\mathrm{left_quotient_map}\left(G,H\right)\right)\left(x\right)=\mathrm{right_coset}\left(G,H,x\right)$
Signatures:
sts

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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

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