AreIsomorphic - Maple Help

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)
 > $H≔\mathrm{SmallGroup}\left(48,29\right)$
 ${H}{≔}{\mathrm{< a permutation group on 48 letters with 5 generators >}}$ (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)
 > $\mathrm{Codomain}\left(\mathrm{iso}\right)$
 ${\mathrm{< a permutation group on 48 letters with 5 generators >}}$ (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)
 > $\mathrm{iso}\left(a·b\right)$
 $\left({1}{,}{7}{,}{9}{,}{8}\right)\left({2}{,}{14}{,}{16}{,}{15}\right)\left({3}{,}{19}{,}{21}{,}{20}\right)\left({4}{,}{24}{,}{26}{,}{25}\right)\left({5}{,}{28}{,}{6}{,}{29}\right)\left({10}{,}{35}{,}{37}{,}{36}\right)\left({11}{,}{40}{,}{42}{,}{41}\right)\left({12}{,}{44}{,}{13}{,}{45}\right)\left({17}{,}{31}{,}{18}{,}{32}\right)\left({22}{,}{30}{,}{23}{,}{27}\right)\left({33}{,}{47}{,}{34}{,}{48}\right)\left({38}{,}{46}{,}{39}{,}{43}\right)$ (12)
 > $\mathrm{iso}\left(a\right)·\mathrm{iso}\left(b\right)$
 $\left({1}{,}{7}{,}{9}{,}{8}\right)\left({2}{,}{14}{,}{16}{,}{15}\right)\left({3}{,}{19}{,}{21}{,}{20}\right)\left({4}{,}{24}{,}{26}{,}{25}\right)\left({5}{,}{28}{,}{6}{,}{29}\right)\left({10}{,}{35}{,}{37}{,}{36}\right)\left({11}{,}{40}{,}{42}{,}{41}\right)\left({12}{,}{44}{,}{13}{,}{45}\right)\left({17}{,}{31}{,}{18}{,}{32}\right)\left({22}{,}{30}{,}{23}{,}{27}\right)\left({33}{,}{47}{,}{34}{,}{48}\right)\left({38}{,}{46}{,}{39}{,}{43}\right)$ (13)
 > $\mathrm{iso}\left(a·b\right)$
 $\left({1}{,}{7}{,}{9}{,}{8}\right)\left({2}{,}{14}{,}{16}{,}{15}\right)\left({3}{,}{19}{,}{21}{,}{20}\right)\left({4}{,}{24}{,}{26}{,}{25}\right)\left({5}{,}{28}{,}{6}{,}{29}\right)\left({10}{,}{35}{,}{37}{,}{36}\right)\left({11}{,}{40}{,}{42}{,}{41}\right)\left({12}{,}{44}{,}{13}{,}{45}\right)\left({17}{,}{31}{,}{18}{,}{32}\right)\left({22}{,}{30}{,}{23}{,}{27}\right)\left({33}{,}{47}{,}{34}{,}{48}\right)\left({38}{,}{46}{,}{39}{,}{43}\right)$ (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)
 > $G≔\mathrm{DirectProduct}\left(A,B\right)$
 ${G}{≔}{{\mathbf{A}}}_{{4}}{×}{{\mathbf{S}}}_{{3}}$ (17)
 > $H≔\mathrm{DirectProduct}\left(B,A\right)$
 ${H}{≔}{{\mathbf{S}}}_{{3}}{×}{{\mathbf{A}}}_{{4}}$ (18)
 > $\mathrm{AreIsomorphic}\left(G,H\right)$
 ${\mathrm{true}}$ (19)

Compatibility

 • The GroupTheory[AreIsomorphic] command was introduced in Maple 17.