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 $\mathrm{G2}$.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$G≔\mathrm{GL}\left(2\,3\right)$

$G≔\mathbf{GL}\left(2\,3\right)$

$H≔\mathrm{SmallGroup}\left(48\,29\right)$

$H≔⟨\left(1\,2\right)\left(3\,11\right)\left(4\,10\right)\left(5\,14\right)\left(6\,15\right)\left(7\,12\right)\left(8\,13\right)\left(9\,16\right)\left(17\,40\right)\left(18\,41\right)\left(19\,38\right)\left(20\,39\right)\left(21\,42\right)\left(22\,35\right)\left(23\,36\right)\left(24\,33\right)\left(25\,34\right)\left(26\,37\right)\left(27\,47\right)\left(28\,45\right)\left(29\,44\right)\left(30\,48\right)\left(31\,43\right)\left(32\,46\right)\,\left(1\,3\,4\right)\left(2\,10\,11\right)\left(5\,20\,27\right)\left(6\,19\,30\right)\left(7\,31\,23\right)\left(8\,32\,22\right)\left(9\,21\,26\right)\left(12\,36\,43\right)\left(13\,35\,46\right)\left(14\,47\,39\right)\left(15\,48\,38\right)\left(16\,37\,42\right)\left(17\,25\,29\right)\left(18\,24\,28\right)\left(33\,41\,45\right)\left(34\,40\,44\right)\,\left(1\,5\,9\,6\right)\left(2\,12\,16\,13\right)\left(3\,17\,21\,18\right)\left(4\,22\,26\,23\right)\left(7\,29\,8\,28\right)\left(10\,33\,37\,34\right)\left(11\,38\,42\,39\right)\left(14\,45\,15\,44\right)\left(19\,32\,20\,31\right)\left(24\,27\,25\,30\right)\left(35\,48\,36\,47\right)\left(40\,43\,41\,46\right)\,\left(1\,7\,9\,8\right)\left(2\,14\,16\,15\right)\left(3\,19\,21\,20\right)\left(4\,24\,26\,25\right)\left(5\,28\,6\,29\right)\left(10\,35\,37\,36\right)\left(11\,40\,42\,41\right)\left(12\,44\,13\,45\right)\left(17\,31\,18\,32\right)\left(22\,30\,23\,27\right)\left(33\,47\,34\,48\right)\left(38\,46\,39\,43\right)\,\left(1\,9\right)\left(2\,16\right)\left(3\,21\right)\left(4\,26\right)\left(5\,6\right)\left(7\,8\right)\left(10\,37\right)\left(11\,42\right)\left(12\,13\right)\left(14\,15\right)\left(17\,18\right)\left(19\,20\right)\left(22\,23\right)\left(24\,25\right)\left(27\,30\right)\left(28\,29\right)\left(31\,32\right)\left(33\,34\right)\left(35\,36\right)\left(38\,39\right)\left(40\,41\right)\left(43\,46\right)\left(44\,45\right)\left(47\,48\right)⟩$

$\mathrm{AreIsomorphic}\left(H\,G\right)$

$\mathrm{true}$

$\mathrm{AreIsomorphic}\left(G\,H\,'\mathrm{assign}=\mathrm{iso}'\right)$

$\mathrm{true}$

$\mathrm{Domain}\left(\mathrm{iso}\right)$

$\mathbf{GL}\left(2\,3\right)$

$\mathrm{Codomain}\left(\mathrm{iso}\right)$

$⟨\left(1\,2\right)\left(3\,11\right)\left(4\,10\right)\left(5\,14\right)\left(6\,15\right)\left(7\,12\right)\left(8\,13\right)\left(9\,16\right)\left(17\,40\right)\left(18\,41\right)\left(19\,38\right)\left(20\,39\right)\left(21\,42\right)\left(22\,35\right)\left(23\,36\right)\left(24\,33\right)\left(25\,34\right)\left(26\,37\right)\left(27\,47\right)\left(28\,45\right)\left(29\,44\right)\left(30\,48\right)\left(31\,43\right)\left(32\,46\right)\,\left(1\,3\,4\right)\left(2\,10\,11\right)\left(5\,20\,27\right)\left(6\,19\,30\right)\left(7\,31\,23\right)\left(8\,32\,22\right)\left(9\,21\,26\right)\left(12\,36\,43\right)\left(13\,35\,46\right)\left(14\,47\,39\right)\left(15\,48\,38\right)\left(16\,37\,42\right)\left(17\,25\,29\right)\left(18\,24\,28\right)\left(33\,41\,45\right)\left(34\,40\,44\right)\,\left(1\,5\,9\,6\right)\left(2\,12\,16\,13\right)\left(3\,17\,21\,18\right)\left(4\,22\,26\,23\right)\left(7\,29\,8\,28\right)\left(10\,33\,37\,34\right)\left(11\,38\,42\,39\right)\left(14\,45\,15\,44\right)\left(19\,32\,20\,31\right)\left(24\,27\,25\,30\right)\left(35\,48\,36\,47\right)\left(40\,43\,41\,46\right)\,\left(1\,7\,9\,8\right)\left(2\,14\,16\,15\right)\left(3\,19\,21\,20\right)\left(4\,24\,26\,25\right)\left(5\,28\,6\,29\right)\left(10\,35\,37\,36\right)\left(11\,40\,42\,41\right)\left(12\,44\,13\,45\right)\left(17\,31\,18\,32\right)\left(22\,30\,23\,27\right)\left(33\,47\,34\,48\right)\left(38\,46\,39\,43\right)\,\left(1\,9\right)\left(2\,16\right)\left(3\,21\right)\left(4\,26\right)\left(5\,6\right)\left(7\,8\right)\left(10\,37\right)\left(11\,42\right)\left(12\,13\right)\left(14\,15\right)\left(17\,18\right)\left(19\,20\right)\left(22\,23\right)\left(24\,25\right)\left(27\,30\right)\left(28\,29\right)\left(31\,32\right)\left(33\,34\right)\left(35\,36\right)\left(38\,39\right)\left(40\,41\right)\left(43\,46\right)\left(44\,45\right)\left(47\,48\right)⟩$

$a≔\mathrm{Perm}\left(\left[\left[1\,6\,2\,3\right]\,\left[4\,7\,8\,5\right]\right]\right)$

$a≔\left(1\,6\,2\,3\right)\left(4\,7\,8\,5\right)$

$b≔\mathrm{Perm}\left(\left[\left[1\,2\right]\,\left[3\,6\right]\,\left[4\,8\right]\,\left[5\,7\right]\right]\right)$

$b≔\left(1\,2\right)\left(3\,6\right)\left(4\,8\right)\left(5\,7\right)$

$a\mathbf{in}G$

$\mathrm{true}$

$b\mathbf{in}G$

$\mathrm{true}$

$a·b$

$\left(1\,3\,2\,6\right)\left(4\,5\,8\,7\right)$

$\mathrm{iso}\left(a·b\right)$

$\left(1\,29\,9\,28\right)\left(2\,45\,16\,44\right)\left(3\,32\,21\,31\right)\left(4\,27\,26\,30\right)\left(5\,7\,6\,8\right)\left(10\,48\,37\,47\right)\left(11\,43\,42\,46\right)\left(12\,14\,13\,15\right)\left(17\,19\,18\,20\right)\left(22\,24\,23\,25\right)\left(33\,35\,34\,36\right)\left(38\,40\,39\,41\right)$

$\mathrm{iso}\left(a\right)·\mathrm{iso}\left(b\right)$

$\left(1\,29\,9\,28\right)\left(2\,45\,16\,44\right)\left(3\,32\,21\,31\right)\left(4\,27\,26\,30\right)\left(5\,7\,6\,8\right)\left(10\,48\,37\,47\right)\left(11\,43\,42\,46\right)\left(12\,14\,13\,15\right)\left(17\,19\,18\,20\right)\left(22\,24\,23\,25\right)\left(33\,35\,34\,36\right)\left(38\,40\,39\,41\right)$

$\mathrm{iso}\left(a·b\right)$

$\left(1\,29\,9\,28\right)\left(2\,45\,16\,44\right)\left(3\,32\,21\,31\right)\left(4\,27\,26\,30\right)\left(5\,7\,6\,8\right)\left(10\,48\,37\,47\right)\left(11\,43\,42\,46\right)\left(12\,14\,13\,15\right)\left(17\,19\,18\,20\right)\left(22\,24\,23\,25\right)\left(33\,35\,34\,36\right)\left(38\,40\,39\,41\right)$

$A≔\mathrm{Alt}\left(4\right)$

$A≔{\mathbf{A}}_4$

$B≔\mathrm{Symm}\left(3\right)$

$B≔{\mathbf{S}}_3$

$G≔\mathrm{DirectProduct}\left(A\,B\right)$

$G≔⟨\left(1\,2\,3\right)\,\left(2\,3\,4\right)\,\left(5\,6\right)\,\left(5\,6\,7\right)⟩$

$H≔\mathrm{DirectProduct}\left(B\,A\right)$

$H≔⟨\left(1\,2\right)\,\left(1\,2\,3\right)\,\left(4\,5\,6\right)\,\left(5\,6\,7\right)⟩$

$\mathrm{AreIsomorphic}\left(G\,H\right)$

$\mathrm{true}$

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

For more information on Maple 17 changes, see Updates in Maple 17.