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:
cyclic_group, dihedral_group, generalized_quaternion_group, 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):
isomorphic ( quaternion_group , quotient_group ( free_group ( a , b ) , normal_closure ( free_group ( a , b ) , apply_to_list ( λ x . expression ( free_group ( a , b ) , x ) , a 4 a 2 b -2 a 2 b a b -1 a -1 ) ) ) )
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


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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):
isomorphic ( dihedral_group ( n ) , quotient_group ( free_group ( a , b ) , normal_closure ( free_group ( a , b ) , apply_to_list ( λ x . expression ( free_group ( a , b ) , x ) , ( a 2 , b n , a b a b ) ) ) ) )
Signatures:
sts


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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


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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


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