The NormalSubgroups( G ) command computes the normal subgroups of a finite group G.

The group G must be an instance of a permutation group or a Cayley table group.

The MinimalNormalSubgroups( G ) command computes the minimal normal subgroups of a permutation group G. These are the non-trivial normal subgroups of G that properly contain no other non-trivial normal subgroup of G.

The MaximalNormalSubgroups( G ) command computes the maximal normal subgroups of a permutation group G. These subgroups are proper normal subgroups of G contained properly in no other proper normal subgroup of G.

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

$G≔\mathrm{Alt}\left(4\right)$

$G≔{\mathbf{A}}_4$

$S≔\mathrm{NormalSubgroups}\left(G\right)$

$S≔\left[⟨\left(1\,4\right)\left(2\,3\right)\,\left(1\,2\right)\left(3\,4\right)\,\left(2\,3\,4\right)⟩\,⟨\left(1\,4\right)\left(2\,3\right)\,\left(1\,2\right)\left(3\,4\right)⟩\,⟨⟩\right]$

$\mathrm{andmap}\left(\mathrm{IsNormal}\,S\,G\right)$

$\mathrm{true}$

$G≔\mathrm{Alt}\left(5\right)$

$G≔{\mathbf{A}}_5$

$\mathrm{NormalSubgroups}\left(G\right)$

$\left[{\mathbf{A}}_5\,⟨⟩\right]$

$G≔\mathrm{DihedralGroup}\left(10\right)$

$G≔{\mathbf{D}}_{10}$

$L≔\mathrm{NormalSubgroups}\left(G\right)$

$L≔\left[⟨\left(1\,6\right)\left(2\,5\right)\left(3\,4\right)\left(7\,10\right)\left(8\,9\right)\,\left(1\,3\right)\left(4\,10\right)\left(5\,9\right)\left(6\,8\right)\,\left(1\,8\right)\left(2\,7\right)\left(3\,6\right)\left(4\,5\right)\left(9\,10\right)⟩\,⟨\left(1\,6\right)\left(2\,5\right)\left(3\,4\right)\left(7\,10\right)\left(8\,9\right)\,\left(1\,8\right)\left(2\,7\right)\left(3\,6\right)\left(4\,5\right)\left(9\,10\right)⟩\,⟨\left(1\,9\,7\,5\,3\right)\left(2\,10\,8\,6\,4\right)\,\left(1\,3\right)\left(4\,10\right)\left(5\,9\right)\left(6\,8\right)⟩\,⟨\left(1\,9\,7\,5\,3\right)\left(2\,10\,8\,6\,4\right)\,\left(1\,10\,9\,8\,7\,6\,5\,4\,3\,2\right)⟩\,⟨\left(1\,9\,7\,5\,3\right)\left(2\,10\,8\,6\,4\right)⟩\,⟨\left(1\,6\right)\left(2\,7\right)\left(3\,8\right)\left(4\,9\right)\left(5\,10\right)⟩\,⟨⟩\right]$

$\mathrm{map}\left(\mathrm{GroupOrder}\,L\right)$

$\left[20\,10\,10\,10\,5\,2\,1\right]$

$\mathrm{map}\left(\mathrm{GroupOrder}\,\mathrm{MinimalNormalSubgroups}\left(G\right)\right)$

$\left[2\,5\right]$

$\mathrm{map}\left(\mathrm{GroupOrder}\,\mathrm{MaximalNormalSubgroups}\left(G\right)\right)$

$\left[10\,10\,10\right]$

$\left(\mathrm{MinimalNormalSubgroups}\,\mathrm{MaximalNormalSubgroups}\right)\left(\mathrm{TrivialGroup}\left(\right)\right)$

$\left[\right],\left[\right]$

$\mathrm{MaximalNormalSubgroups}\left(\mathrm{Suzuki2B2}\left(32\right)\right)$

$\mathrm{MinimalNormalSubgroups}\left(\mathrm{McLaughlinGroup}\left(\right)\right)$

$\left[McL\right]$

$\mathbf{use}\mathrm{GraphTheory}\mathbf{in}A≔\mathrm{AutomorphismGroup}\left(\mathrm{SpecialGraphs}:-\mathrm{ClebschGraph}\left(\right)\right)\mathbf{end\; use}$

$⟨\left(4\,13\right)\left(5\,6\right)\left(8\,14\right)\left(9\,11\right)\,\left(2\,4\right)\left(5\,16\right)\left(11\,15\right)\left(12\,14\right)\,\left(1\,2\,5\,3\,9\,11\,12\,8\right)\left(4\,14\,6\,13\,15\,16\,7\,10\right)\,\left(3\,7\right)\left(5\,14\right)\left(6\,8\right)\left(12\,16\right)\,\left(2\,5\right)\left(4\,16\right)\left(7\,9\right)\left(8\,10\right)⟩$

$\mathrm{GroupOrder}\left(A\right)$

$1920$

$\mathrm{NA}≔\mathrm{NormalSubgroups}\left(A\right)$

$\mathrm{NA}≔\left[⟨\left(1\,3\,8\,13\,11\right)\left(2\,7\,12\,16\,6\right)\left(5\,15\,14\,9\,10\right)\,\left(1\,3\,12\,6\,13\,11\right)\left(2\,7\,8\right)\left(5\,9\,10\right)\left(14\,15\right)\,\left(1\,3\,14\,4\,9\right)\left(2\,7\,12\,16\,5\right)\left(6\,15\,8\,11\,10\right)⟩\,⟨\left(1\,3\,8\,13\,11\right)\left(2\,7\,12\,16\,6\right)\left(5\,15\,14\,9\,10\right)\,\left(1\,3\,14\,4\,9\right)\left(2\,7\,12\,16\,5\right)\left(6\,15\,8\,11\,10\right)⟩\,⟨\left(1\,5\right)\left(2\,6\right)\left(3\,4\right)\left(7\,11\right)\left(8\,15\right)\left(9\,12\right)\left(10\,14\right)\left(13\,16\right)\,\left(1\,6\right)\left(2\,5\right)\left(3\,13\right)\left(4\,16\right)\left(7\,9\right)\left(8\,10\right)\left(11\,12\right)\left(14\,15\right)\,\left(1\,14\right)\left(2\,8\right)\left(3\,11\right)\left(4\,7\right)\left(5\,10\right)\left(6\,15\right)\left(9\,16\right)\left(12\,13\right)\,\left(1\,16\right)\left(2\,3\right)\left(4\,6\right)\left(5\,13\right)\left(7\,15\right)\left(8\,11\right)\left(9\,14\right)\left(10\,12\right)⟩\,⟨⟩\right]$

$\mathrm{GroupOrder}\left(\mathrm{NA}\left[2\right]\right)$

$960$

$\mathrm{IsPerfect}\left(\mathrm{NA}\left[2\right]\right)$

$\mathrm{true}$

$\mathrm{IsPrimitive}\left(\mathrm{NA}\left[2\right]\right)$

$\mathrm{true}$

$L≔\mathrm{NormalSubgroups}\left(\mathrm{CyclicGroup}\left(36\right)\right)$

$L≔\left[⟨\left(1\,2\,3\,4\,5\,6\,7\,8\,9\,10\,11\,12\,13\,14\,15\,16\,17\,18\,19\,20\,21\,22\,23\,24\,25\,26\,27\,28\,29\,30\,31\,32\,33\,34\,35\,36\right)⟩\,⟨\left(1\,3\,5\,7\,9\,11\,13\,15\,17\,19\,21\,23\,25\,27\,29\,31\,33\,35\right)\left(2\,4\,6\,8\,10\,12\,14\,16\,18\,20\,22\,24\,26\,28\,30\,32\,34\,36\right)⟩\,⟨\left(1\,4\,7\,10\,13\,16\,19\,22\,25\,28\,31\,34\right)\left(2\,5\,8\,11\,14\,17\,20\,23\,26\,29\,32\,35\right)\left(3\,6\,9\,12\,15\,18\,21\,24\,27\,30\,33\,36\right)⟩\,⟨\left(1\,5\,9\,13\,17\,21\,25\,29\,33\right)\left(2\,6\,10\,14\,18\,22\,26\,30\,34\right)\left(3\,7\,11\,15\,19\,23\,27\,31\,35\right)\left(4\,8\,12\,16\,20\,24\,28\,32\,36\right)⟩\,⟨\left(1\,7\,13\,19\,25\,31\right)\left(2\,8\,14\,20\,26\,32\right)\left(3\,9\,15\,21\,27\,33\right)\left(4\,10\,16\,22\,28\,34\right)\left(5\,11\,17\,23\,29\,35\right)\left(6\,12\,18\,24\,30\,36\right)⟩\,⟨\left(1\,10\,19\,28\right)\left(2\,11\,20\,29\right)\left(3\,12\,21\,30\right)\left(4\,13\,22\,31\right)\left(5\,14\,23\,32\right)\left(6\,15\,24\,33\right)\left(7\,16\,25\,34\right)\left(8\,17\,26\,35\right)\left(9\,18\,27\,36\right)⟩\,⟨\left(1\,13\,25\right)\left(2\,14\,26\right)\left(3\,15\,27\right)\left(4\,16\,28\right)\left(5\,17\,29\right)\left(6\,18\,30\right)\left(7\,19\,31\right)\left(8\,20\,32\right)\left(9\,21\,33\right)\left(10\,22\,34\right)\left(11\,23\,35\right)\left(12\,24\,36\right)⟩\,⟨\left(1\,19\right)\left(2\,20\right)\left(3\,21\right)\left(4\,22\right)\left(5\,23\right)\left(6\,24\right)\left(7\,25\right)\left(8\,26\right)\left(9\,27\right)\left(10\,28\right)\left(11\,29\right)\left(12\,30\right)\left(13\,31\right)\left(14\,32\right)\left(15\,33\right)\left(16\,34\right)\left(17\,35\right)\left(18\,36\right)⟩\,⟨⟩\right]$

$L≔\mathrm{MaximalNormalSubgroups}\left(\mathrm{CyclicGroup}\left(100000\right)\right)\:$

$\mathrm{map}\left(\mathrm{GroupOrder}\,L\right)$

$\left[50000\,20000\right]$

$L≔\mathrm{MinimalNormalSubgroups}\left(\mathrm{CyclicGroup}\left(100000\right)\right)\:$

$\mathrm{map}\left(\mathrm{GroupOrder}\,L\right)$

$\left[5\,2\right]$

$G≔\mathrm{CayleyTableGroup}\left(\mathrm{Symm}\left(3\right)\right)$

$G≔\mathrm{<\; a\; Cayley\; table\; group\; with\; 6\; elements\; >}$

$\mathrm{NormalSubgroups}\left(G\right)$

$\left[\mathrm{<\; a\; Cayley\; table\; group\; with\; 1\; element\; >}\&comma;\mathrm{<\; a\; Cayley\; table\; group\; with\; 3\; elements\; >}\&comma;\mathrm{<\; a\; Cayley\; table\; group\; with\; 6\; elements\; >}\right]$

The GroupTheory[NormalSubgroups] command was introduced in Maple 2016.

For more information on Maple 2016 changes, see Updates in Maple 2016.