AllSmallGroups - Maple Help

GroupTheory

 AllSmallGroups

 Calling Sequence AllSmallGroups( r ) AllSmallGroups( r, f )

Parameters

 r - a positive integer, an integer range, or a small group ID range f - optional equation: form=permgroup (default) or form=fpgroup

Description

 • The small groups library contains all groups of small orders up to $511$. The groups are sorted by their orders and they are listed up to isomorphism; that is, for each of the available orders a complete and irredundant list of isomorphism type representatives of groups is given. These groups are available as permutation groups and as groups defined by generators and relations.
 • In its simplest form, the command AllSmallGroups( r ) returns a list of all the small groups in the small groups library of order r, where r is a positive integer less than $512$.
 • If r is a range of the form m .. n, where m and n are positive integers, then AllSmallGroups( r ) returns a list of all the groups whose order lies in the range m .. n.
 • More generally, r may be a "range" of the form [ln, lk] .. [un, uk], where ln and un are positive integer less than $512$, and where lk is a positive integer in the range 1 .. NumGroups( ln ), and uk is an integer in the range 1 .. NumGroups( un ). In this case, AllSmallGroups( r ) returns a list of the groups whose orders lie in the range ln .. un, beginning with the lk-th group of order ln, and ending with the uk-th group of order un. Think of the groups of each order as forming a "row" of a "ragged" matrix, and the first operand of the range r specifies a first position  in this matrix, while the second operand of r specifies a second position in the matrix, so that the range r selects all the groups occurring between these two positions, where the matrix is traversed in row-major order, from the first to the second position.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $\mathrm{AllSmallGroups}\left(6\right)$
 $\left[⟨\left({1}{,}{2}\right)\left({3}{,}{6}\right)\left({4}{,}{5}\right){,}\left({1}{,}{3}{,}{4}\right)\left({2}{,}{5}{,}{6}\right)⟩{,}⟨\left({1}{,}{2}{,}{4}{,}{6}{,}{5}{,}{3}\right)⟩\right]$ (1)
 > $\mathrm{AllSmallGroups}\left(6,\mathrm{form}="fpgroup"\right)$
 $\left[⟨{}{\mathrm{_a}}{,}{\mathrm{_b}}{}{\mid }{}{{\mathrm{_a}}}^{{2}}{,}{{\mathrm{_b}}}^{{3}}{,}{{\mathrm{_a}}}^{{-1}}{}{\mathrm{_b}}{}{\mathrm{_a}}{}{{\mathrm{_b}}}^{{-2}}{}⟩{,}⟨{}{\mathrm{g0}}{}{\mid }{}{{\mathrm{g0}}}^{{6}}{}⟩\right]$ (2)
 > $\mathrm{AllSmallGroups}\left(6,\mathrm{form}="permgroup"\right)$
 $\left[⟨\left({1}{,}{2}\right)\left({3}{,}{6}\right)\left({4}{,}{5}\right){,}\left({1}{,}{3}{,}{4}\right)\left({2}{,}{5}{,}{6}\right)⟩{,}⟨\left({1}{,}{2}{,}{4}{,}{6}{,}{5}{,}{3}\right)⟩\right]$ (3)
 > $\mathrm{AllSmallGroups}\left(4..6\right)$
 $\left[⟨\left({1}{,}{2}{,}{4}{,}{3}\right)⟩{,}⟨\left({1}{,}{2}\right)\left({3}{,}{4}\right){,}\left({1}{,}{3}\right)\left({2}{,}{4}\right)⟩{,}⟨\left({1}{,}{2}{,}{4}{,}{5}{,}{3}\right)⟩{,}⟨\left({1}{,}{2}\right)\left({3}{,}{6}\right)\left({4}{,}{5}\right){,}\left({1}{,}{3}{,}{4}\right)\left({2}{,}{5}{,}{6}\right)⟩{,}⟨\left({1}{,}{2}{,}{4}{,}{6}{,}{5}{,}{3}\right)⟩\right]$ (4)
 > $\mathrm{AllSmallGroups}\left(\left[6,2\right]..\left[8,3\right]\right)$
 $\left[⟨\left({1}{,}{2}{,}{4}{,}{6}{,}{5}{,}{3}\right)⟩{,}⟨\left({1}{,}{2}{,}{4}{,}{6}{,}{7}{,}{5}{,}{3}\right)⟩{,}⟨\left({1}{,}{2}{,}{4}{,}{6}{,}{8}{,}{7}{,}{5}{,}{3}\right)⟩{,}⟨\left({1}{,}{2}{,}{5}{,}{3}\right)\left({4}{,}{6}{,}{8}{,}{7}\right){,}\left({1}{,}{4}\right)\left({2}{,}{6}\right)\left({3}{,}{7}\right)\left({5}{,}{8}\right)⟩{,}⟨\left({1}{,}{2}\right)\left({3}{,}{7}\right)\left({4}{,}{6}\right)\left({5}{,}{8}\right){,}\left({1}{,}{3}\right)\left({2}{,}{5}\right)\left({4}{,}{8}\right)\left({6}{,}{7}\right){,}\left({1}{,}{4}\right)\left({2}{,}{6}\right)\left({3}{,}{8}\right)\left({5}{,}{7}\right)⟩\right]$ (5)
 > $\mathrm{AllSmallGroups}\left(\left[6,2\right]..\left[8,3\right],'\mathrm{form}'="fpgroup"\right)$
 $\left[⟨{}{\mathrm{g7}}{}{\mid }{}{{\mathrm{g7}}}^{{6}}{}⟩{,}⟨{}{\mathrm{g8}}{}{\mid }{}{{\mathrm{g8}}}^{{7}}{}⟩{,}⟨{}{\mathrm{_a}}{}{\mid }{}{{\mathrm{_a}}}^{{8}}{}⟩{,}⟨{}{\mathrm{_a}}{,}{\mathrm{_b}}{}{\mid }{}{{\mathrm{_b}}}^{{2}}{,}{{\mathrm{_a}}}^{{4}}{,}{{\mathrm{_a}}}^{{-1}}{}{{\mathrm{_b}}}^{{-1}}{}{\mathrm{_a}}{}{\mathrm{_b}}{}⟩{,}⟨{}{\mathrm{_a}}{,}{\mathrm{_b}}{,}{\mathrm{_c}}{}{\mid }{}{{\mathrm{_a}}}^{{2}}{,}{{\mathrm{_b}}}^{{2}}{,}{{\mathrm{_c}}}^{{2}}{,}{{\mathrm{_c}}}^{{-1}}{}{{\mathrm{_a}}}^{{-1}}{}{\mathrm{_c}}{}{\mathrm{_a}}{,}{{\mathrm{_c}}}^{{-1}}{}{{\mathrm{_b}}}^{{-1}}{}{\mathrm{_c}}{}{\mathrm{_b}}{,}{{\mathrm{_b}}}^{{-1}}{}{{\mathrm{_a}}}^{{-1}}{}{\mathrm{_b}}{}{\mathrm{_a}}{}{{\mathrm{_c}}}^{{-1}}{}⟩\right]$ (6)

Compatibility

 • The GroupTheory[AllSmallGroups] command was introduced in Maple 17.
 • For more information on Maple 17 changes, see Updates in Maple 17.