Difference between revisions of "Group Examples"
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=== Noncommutative Groups === | === Noncommutative Groups === | ||
| + | |||
| + | ==== Baumslag groups ==== | ||
| + | The [ftp://apcocoa.org/pub/symbolic_data/non_commutative/baumslag.coc Baumslag] (respectively Baumslag-Solitar) groups are examples of two-generator one-relator groups. | ||
| + | The first variante of this group has the presentation <a, b | a^m = b^n = 1 > for m, n natural numbers. | ||
| + | Type Baumslag1(m, n, [DegreeBound, LoopBound]) to calculate the Gröbner base. | ||
| + | The second variante of this group (the Baumslag-Solitar group) has the presentation <a, b | b*a^m = a^n*b> for m, n natural numbers. | ||
| + | Type Baumslag2(m, n, [DegreeBound, LoopBound]) to calculate the Gröbner base. | ||
| + | |||
| + | ==== Braid groups ==== | ||
| + | For a natural number n the [ftp://apcocoa.org/pub/symbolic_data/non_commutative/braid.coc Braid] group has n strands and n - 1 generators: g_1, g_2, ... , g_(n - 1) and relations: | ||
| + | g_i * g_j = g_j * g_i for |i - j| >= 2 and | ||
| + | g_i * g_(i + 1) * g_i = g_(i + 1) * g_i * g_(i + 1) for 1 <= i <= n - 2 | ||
| + | Type Braid(n, [DegreeBound, LoopBound]) to calculate the Gröbner base. | ||
| + | |||
| + | ==== Dihedral groups ==== | ||
| + | The [ftp://apcocoa.org/pub/symbolic_data/non_commutative/dihedral.coc Dihedral] group of degree n (natural number) has the presentation <a, b | a^2 = b^n = 1, aba = b^-1> | ||
| + | Type Dihedral(n, [DegreeBound, LoopBound]) to calculate the Gröbner base. | ||
Revision as of 13:32, 16 August 2012
ftp://apcocoa.org/pub/symbolic_data
Noncommutative Groups
Baumslag groups
The Baumslag (respectively Baumslag-Solitar) groups are examples of two-generator one-relator groups. The first variante of this group has the presentation <a, b | a^m = b^n = 1 > for m, n natural numbers. Type Baumslag1(m, n, [DegreeBound, LoopBound]) to calculate the Gröbner base. The second variante of this group (the Baumslag-Solitar group) has the presentation <a, b | b*a^m = a^n*b> for m, n natural numbers. Type Baumslag2(m, n, [DegreeBound, LoopBound]) to calculate the Gröbner base.
Braid groups
For a natural number n the Braid group has n strands and n - 1 generators: g_1, g_2, ... , g_(n - 1) and relations: g_i * g_j = g_j * g_i for |i - j| >= 2 and g_i * g_(i + 1) * g_i = g_(i + 1) * g_i * g_(i + 1) for 1 <= i <= n - 2 Type Braid(n, [DegreeBound, LoopBound]) to calculate the Gröbner base.
Dihedral groups
The Dihedral group of degree n (natural number) has the presentation <a, b | a^2 = b^n = 1, aba = b^-1> Type Dihedral(n, [DegreeBound, LoopBound]) to calculate the Gröbner base.
