AreIsomorphic - Maple Help
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GroupTheory

 AreIsomorphic
 test if two groups are isomorphic

 Calling Sequence AreIsomorphic(G1, G2) AreIsomorphic(G1, G2, assign = iso) iso(g1) Domain(iso) Codomain(iso)

Parameters

 G1, G2 - groups iso - mapping returned by AreIsomorphic g1 - element of G1

Description

 • The AreIsomorphic command tests if two groups are isomorphic. It returns true if they are and false if they are not.
 • If G1 and G2 are indeed isomorphic, then Maple will eventually attempt to construct an isomorphism. You can have this isomorphism assigned to a variable name by using the assign option: if you specify $'\mathrm{assign}'='\mathrm{iso}'$, then the isomorphism will be assigned to the variable name iso. This variable can then function as a procedure (or more precisely, a module with ModuleApply) mapping elements from $\mathrm{G1}$ to $\mathrm{G2}$. Concretely, if $g\in \mathrm{G1}$, and we have specified $'\mathrm{assign}'='\mathrm{iso}'$ then the call $\mathrm{iso}\left(g\right)$ will return the element of $\mathrm{G2}$ corresponding to $g$.
 • An isomorphism object assigned by AreIsomorphic can be interrogated about its domain and codomain using the Domain and Codomain procedures. If iso was assigned by a call $\mathrm{AreIsomorphic}\left(\mathrm{G1},\mathrm{G2},'\mathrm{assign}'='\mathrm{iso}'\right)$, then $\mathrm{Domain}\left(\mathrm{iso}\right)$ returns $\mathrm{G1}$ and $\mathrm{Codomain}\left(\mathrm{iso}\right)$ returns $\mathrm{G2}$.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{GL}\left(2,3\right)$
 ${G}{≔}{\mathbf{GL}}\left({2}{,}{3}\right)$ (1)
 > $H≔\mathrm{SmallGroup}\left(48,29\right)$
 ${H}{≔}⟨\left({1}{,}{2}\right)\left({3}{,}{11}\right)\left({4}{,}{10}\right)\left({5}{,}{14}\right)\left({6}{,}{15}\right)\left({7}{,}{12}\right)\left({8}{,}{13}\right)\left({9}{,}{16}\right)\left({17}{,}{40}\right)\left({18}{,}{41}\right)\left({19}{,}{38}\right)\left({20}{,}{39}\right)\left({21}{,}{42}\right)\left({22}{,}{35}\right)\left({23}{,}{36}\right)\left({24}{,}{33}\right)\left({25}{,}{34}\right)\left({26}{,}{37}\right)\left({27}{,}{47}\right)\left({28}{,}{45}\right)\left({29}{,}{44}\right)\left({30}{,}{48}\right)\left({31}{,}{43}\right)\left({32}{,}{46}\right){,}\left({1}{,}{3}{,}{4}\right)\left({2}{,}{10}{,}{11}\right)\left({5}{,}{20}{,}{27}\right)\left({6}{,}{19}{,}{30}\right)\left({7}{,}{31}{,}{23}\right)\left({8}{,}{32}{,}{22}\right)\left({9}{,}{21}{,}{26}\right)\left({12}{,}{36}{,}{43}\right)\left({13}{,}{35}{,}{46}\right)\left({14}{,}{47}{,}{39}\right)\left({15}{,}{48}{,}{38}\right)\left({16}{,}{37}{,}{42}\right)\left({17}{,}{25}{,}{29}\right)\left({18}{,}{24}{,}{28}\right)\left({33}{,}{41}{,}{45}\right)\left({34}{,}{40}{,}{44}\right){,}\left({1}{,}{5}{,}{9}{,}{6}\right)\left({2}{,}{12}{,}{16}{,}{13}\right)\left({3}{,}{17}{,}{21}{,}{18}\right)\left({4}{,}{22}{,}{26}{,}{23}\right)\left({7}{,}{29}{,}{8}{,}{28}\right)\left({10}{,}{33}{,}{37}{,}{34}\right)\left({11}{,}{38}{,}{42}{,}{39}\right)\left({14}{,}{45}{,}{15}{,}{44}\right)\left({19}{,}{32}{,}{20}{,}{31}\right)\left({24}{,}{27}{,}{25}{,}{30}\right)\left({35}{,}{48}{,}{36}{,}{47}\right)\left({40}{,}{43}{,}{41}{,}{46}\right){,}\left({1}{,}{7}{,}{9}{,}{8}\right)\left({2}{,}{14}{,}{16}{,}{15}\right)\left({3}{,}{19}{,}{21}{,}{20}\right)\left({4}{,}{24}{,}{26}{,}{25}\right)\left({5}{,}{28}{,}{6}{,}{29}\right)\left({10}{,}{35}{,}{37}{,}{36}\right)\left({11}{,}{40}{,}{42}{,}{41}\right)\left({12}{,}{44}{,}{13}{,}{45}\right)\left({17}{,}{31}{,}{18}{,}{32}\right)\left({22}{,}{30}{,}{23}{,}{27}\right)\left({33}{,}{47}{,}{34}{,}{48}\right)\left({38}{,}{46}{,}{39}{,}{43}\right){,}\left({1}{,}{9}\right)\left({2}{,}{16}\right)\left({3}{,}{21}\right)\left({4}{,}{26}\right)\left({5}{,}{6}\right)\left({7}{,}{8}\right)\left({10}{,}{37}\right)\left({11}{,}{42}\right)\left({12}{,}{13}\right)\left({14}{,}{15}\right)\left({17}{,}{18}\right)\left({19}{,}{20}\right)\left({22}{,}{23}\right)\left({24}{,}{25}\right)\left({27}{,}{30}\right)\left({28}{,}{29}\right)\left({31}{,}{32}\right)\left({33}{,}{34}\right)\left({35}{,}{36}\right)\left({38}{,}{39}\right)\left({40}{,}{41}\right)\left({43}{,}{46}\right)\left({44}{,}{45}\right)\left({47}{,}{48}\right)⟩$ (2)
 > $\mathrm{AreIsomorphic}\left(H,G\right)$
 ${\mathrm{true}}$ (3)
 > $\mathrm{AreIsomorphic}\left(G,H,'\mathrm{assign}=\mathrm{iso}'\right)$
 ${\mathrm{true}}$ (4)
 > $\mathrm{Domain}\left(\mathrm{iso}\right)$
 ${\mathbf{GL}}\left({2}{,}{3}\right)$ (5)
 > $\mathrm{Codomain}\left(\mathrm{iso}\right)$
 $⟨\left({1}{,}{2}\right)\left({3}{,}{11}\right)\left({4}{,}{10}\right)\left({5}{,}{14}\right)\left({6}{,}{15}\right)\left({7}{,}{12}\right)\left({8}{,}{13}\right)\left({9}{,}{16}\right)\left({17}{,}{40}\right)\left({18}{,}{41}\right)\left({19}{,}{38}\right)\left({20}{,}{39}\right)\left({21}{,}{42}\right)\left({22}{,}{35}\right)\left({23}{,}{36}\right)\left({24}{,}{33}\right)\left({25}{,}{34}\right)\left({26}{,}{37}\right)\left({27}{,}{47}\right)\left({28}{,}{45}\right)\left({29}{,}{44}\right)\left({30}{,}{48}\right)\left({31}{,}{43}\right)\left({32}{,}{46}\right){,}\left({1}{,}{3}{,}{4}\right)\left({2}{,}{10}{,}{11}\right)\left({5}{,}{20}{,}{27}\right)\left({6}{,}{19}{,}{30}\right)\left({7}{,}{31}{,}{23}\right)\left({8}{,}{32}{,}{22}\right)\left({9}{,}{21}{,}{26}\right)\left({12}{,}{36}{,}{43}\right)\left({13}{,}{35}{,}{46}\right)\left({14}{,}{47}{,}{39}\right)\left({15}{,}{48}{,}{38}\right)\left({16}{,}{37}{,}{42}\right)\left({17}{,}{25}{,}{29}\right)\left({18}{,}{24}{,}{28}\right)\left({33}{,}{41}{,}{45}\right)\left({34}{,}{40}{,}{44}\right){,}\left({1}{,}{5}{,}{9}{,}{6}\right)\left({2}{,}{12}{,}{16}{,}{13}\right)\left({3}{,}{17}{,}{21}{,}{18}\right)\left({4}{,}{22}{,}{26}{,}{23}\right)\left({7}{,}{29}{,}{8}{,}{28}\right)\left({10}{,}{33}{,}{37}{,}{34}\right)\left({11}{,}{38}{,}{42}{,}{39}\right)\left({14}{,}{45}{,}{15}{,}{44}\right)\left({19}{,}{32}{,}{20}{,}{31}\right)\left({24}{,}{27}{,}{25}{,}{30}\right)\left({35}{,}{48}{,}{36}{,}{47}\right)\left({40}{,}{43}{,}{41}{,}{46}\right){,}\left({1}{,}{7}{,}{9}{,}{8}\right)\left({2}{,}{14}{,}{16}{,}{15}\right)\left({3}{,}{19}{,}{21}{,}{20}\right)\left({4}{,}{24}{,}{26}{,}{25}\right)\left({5}{,}{28}{,}{6}{,}{29}\right)\left({10}{,}{35}{,}{37}{,}{36}\right)\left({11}{,}{40}{,}{42}{,}{41}\right)\left({12}{,}{44}{,}{13}{,}{45}\right)\left({17}{,}{31}{,}{18}{,}{32}\right)\left({22}{,}{30}{,}{23}{,}{27}\right)\left({33}{,}{47}{,}{34}{,}{48}\right)\left({38}{,}{46}{,}{39}{,}{43}\right){,}\left({1}{,}{9}\right)\left({2}{,}{16}\right)\left({3}{,}{21}\right)\left({4}{,}{26}\right)\left({5}{,}{6}\right)\left({7}{,}{8}\right)\left({10}{,}{37}\right)\left({11}{,}{42}\right)\left({12}{,}{13}\right)\left({14}{,}{15}\right)\left({17}{,}{18}\right)\left({19}{,}{20}\right)\left({22}{,}{23}\right)\left({24}{,}{25}\right)\left({27}{,}{30}\right)\left({28}{,}{29}\right)\left({31}{,}{32}\right)\left({33}{,}{34}\right)\left({35}{,}{36}\right)\left({38}{,}{39}\right)\left({40}{,}{41}\right)\left({43}{,}{46}\right)\left({44}{,}{45}\right)\left({47}{,}{48}\right)⟩$ (6)
 > $a≔\mathrm{Perm}\left(\left[\left[1,6,2,3\right],\left[4,7,8,5\right]\right]\right)$
 ${a}{≔}\left({1}{,}{6}{,}{2}{,}{3}\right)\left({4}{,}{7}{,}{8}{,}{5}\right)$ (7)
 > $b≔\mathrm{Perm}\left(\left[\left[1,2\right],\left[3,6\right],\left[4,8\right],\left[5,7\right]\right]\right)$
 ${b}{≔}\left({1}{,}{2}\right)\left({3}{,}{6}\right)\left({4}{,}{8}\right)\left({5}{,}{7}\right)$ (8)
 > $a\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}G$
 ${\mathrm{true}}$ (9)
 > $b\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}G$
 ${\mathrm{true}}$ (10)
 > $a·b$
 $\left({1}{,}{3}{,}{2}{,}{6}\right)\left({4}{,}{5}{,}{8}{,}{7}\right)$ (11)
 > $\mathrm{iso}\left(a·b\right)$
 $\left({1}{,}{29}{,}{9}{,}{28}\right)\left({2}{,}{45}{,}{16}{,}{44}\right)\left({3}{,}{32}{,}{21}{,}{31}\right)\left({4}{,}{27}{,}{26}{,}{30}\right)\left({5}{,}{7}{,}{6}{,}{8}\right)\left({10}{,}{48}{,}{37}{,}{47}\right)\left({11}{,}{43}{,}{42}{,}{46}\right)\left({12}{,}{14}{,}{13}{,}{15}\right)\left({17}{,}{19}{,}{18}{,}{20}\right)\left({22}{,}{24}{,}{23}{,}{25}\right)\left({33}{,}{35}{,}{34}{,}{36}\right)\left({38}{,}{40}{,}{39}{,}{41}\right)$ (12)
 > $\mathrm{iso}\left(a\right)·\mathrm{iso}\left(b\right)$
 $\left({1}{,}{29}{,}{9}{,}{28}\right)\left({2}{,}{45}{,}{16}{,}{44}\right)\left({3}{,}{32}{,}{21}{,}{31}\right)\left({4}{,}{27}{,}{26}{,}{30}\right)\left({5}{,}{7}{,}{6}{,}{8}\right)\left({10}{,}{48}{,}{37}{,}{47}\right)\left({11}{,}{43}{,}{42}{,}{46}\right)\left({12}{,}{14}{,}{13}{,}{15}\right)\left({17}{,}{19}{,}{18}{,}{20}\right)\left({22}{,}{24}{,}{23}{,}{25}\right)\left({33}{,}{35}{,}{34}{,}{36}\right)\left({38}{,}{40}{,}{39}{,}{41}\right)$ (13)
 > $\mathrm{iso}\left(a·b\right)$
 $\left({1}{,}{29}{,}{9}{,}{28}\right)\left({2}{,}{45}{,}{16}{,}{44}\right)\left({3}{,}{32}{,}{21}{,}{31}\right)\left({4}{,}{27}{,}{26}{,}{30}\right)\left({5}{,}{7}{,}{6}{,}{8}\right)\left({10}{,}{48}{,}{37}{,}{47}\right)\left({11}{,}{43}{,}{42}{,}{46}\right)\left({12}{,}{14}{,}{13}{,}{15}\right)\left({17}{,}{19}{,}{18}{,}{20}\right)\left({22}{,}{24}{,}{23}{,}{25}\right)\left({33}{,}{35}{,}{34}{,}{36}\right)\left({38}{,}{40}{,}{39}{,}{41}\right)$ (14)

This example demonstrates that the direct product construction is commutative up to isomorphism.

 > $A≔\mathrm{Alt}\left(4\right)$
 ${A}{≔}{{\mathbf{A}}}_{{4}}$ (15)
 > $B≔\mathrm{Symm}\left(3\right)$
 ${B}{≔}{{\mathbf{S}}}_{{3}}$ (16)
 > $G≔\mathrm{DirectProduct}\left(A,B\right)$
 ${G}{≔}⟨\left({1}{,}{2}{,}{3}\right){,}\left({2}{,}{3}{,}{4}\right){,}\left({5}{,}{6}\right){,}\left({5}{,}{6}{,}{7}\right)⟩$ (17)
 > $H≔\mathrm{DirectProduct}\left(B,A\right)$
 ${H}{≔}⟨\left({1}{,}{2}\right){,}\left({1}{,}{2}{,}{3}\right){,}\left({4}{,}{5}{,}{6}\right){,}\left({5}{,}{6}{,}{7}\right)⟩$ (18)
 > $\mathrm{AreIsomorphic}\left(G,H\right)$
 ${\mathrm{true}}$ (19)

Compatibility

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