# 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):
$\mathrm{isomorphic}\left(\mathrm{quaternion_group},\mathrm{quotient_group}\left(\mathrm{free_group}\left(a,b\right),\mathrm{normal_closure}\left(\mathrm{free_group}\left(a,b\right),\mathrm{apply_to_list}\left(\lambda x.\mathrm{expression}\left(\mathrm{free_group}\left(a,b\right),x\right),\begin{array}{c}{a}^{4}\hfill \\ {a}^{2}{b}^{-2}\hfill \\ {a}^{2}ba{b}^{-1}{a}^{-1}\hfill \end{array}\right)\right)\right)\right)$
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):
$\mathrm{isomorphic}\left(\mathrm{dihedral_group}\left(n\right),\mathrm{quotient_group}\left(\mathrm{free_group}\left(a,b\right),\mathrm{normal_closure}\left(\mathrm{free_group}\left(a,b\right),\mathrm{apply_to_list}\left(\lambda x.\mathrm{expression}\left(\mathrm{free_group}\left(a,b\right),x\right),\left({a}^{2},{b}^{n},abab\right)\right)\right)\right)\right)$
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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