directed = true or false

If the option directed = true or directed is specified, then every edge in the subgroup lattice graph is directed from the smaller to the larger subgroup. If directed = false is specified (the default) then this is not the case.

labels = "integers" (the default) or labels = "ids" or labels = "none"

By default, each vertex is labeled with a single integer: the position of the corresponding subgroup in the list of subgroups obtained by calling convert( L, list ). This behavior is explicitly specified by including the option labels = "integers".

If you specify the option labels = "none", the vertices will not be labeled. This is useful for very large lattices whose structure could be obscured by vertex labels.

The labels = "ids" option causes each vertex to be labeled with the small group identifier for normal subgroup the vertex represents. This is determined by the IdentifySmallGroup command, which returns a pair of the form n, k, where n is the order of the group, and k is an index identifying which group of order n it is. The vertex is then labeled with the expression n/k.

indices or indices = true or indices = false

If the option indices = true or indices is specified, then every edge in the subgroup lattice graph, corresponding, say, to an inclusion of a normal subgroup N into a subgroup K, is labeled with the index of N in K. If indices = false is specified (the default) there are no labels on edges.

vertexcolor = color and edgecolor = color

These two options control the color for subgroups not specified by any of the following options, and the color with which the edges are colored, respectively. The default is white for the vertex color and gray for the edge color. The colors need to be specified in a way that is recognized by ColorTools[Color]. Maple also accepts the alternative spellings vertexcolour and edgecolour.

highlight = subgroup, NormalSeries, or a list of subgroups and NormalSeries

Use the highlight option to visually highlight selected subgroups in the normal subgroup lattice by displaying them in a particular color. There are a several variations on how highlighted subgroups can be specified.

If you just specify a subgroup, or a list of subgroups, then the vertices in the display are highlighted using the default highlight color, which is "red".

Alternatively, you can specify an equation of the form H = color, where H is some normal subgroup in the lattice and color is a color specification understood by the ColorTools[Color] command.

To highlight more than one subgroup or normal series, enclose the equations in a list, such as [ N = color1, K = color2 ].

You can also use a list of subgroups, such as [H, N, K] = color, where H, N and K are all normal subgroups in the lattice. This will cause all the subgroups in the list to be highlighted using the same color color.

If S is a NormalSeries object, you can use S = color to highlight the members of S with the chosen color color. This is equivalent to [seq]( H = color, H = S ).

Subgroups in the normal subgroup lattice not mentioned using the highlight option are displayed using the vertex color, either the default, or the color specified by using the vertexcolor option.

title = expr

The given expression is typeset as the title above the subgroup lattice is specified.

titlefont = list

The title is printed with this font.  The font is specified as explained in plot/options.

The DrawNormalSubgroupLattice( G ) command displays a plot that shows the structure of the lattice of normal subgroups of the group G. The group G must be a group for which it is possible to compute the normal subgroups. In particular, G must be a finite group.

The lattice is displayed as a graph with a vertex for each normal subgroup of G, and with an edge directed from a normal subgroup N to a normal subgroup K if N is contained in K, and there are no normal subgroups properly between N and K. That is, the graph represents the covering relation for the normal subgroup lattice.

The DrawNormalSubgroupLattice command takes a number of options, described below, that you can use to customize the way the lattice is displayed.

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

$G≔\mathrm{QuaternionGroup}\left(\right)$

$G≔Q$

$\mathrm{P1}≔\mathrm{DrawNormalSubgroupLattice}\left(G\,'\mathrm{title}'=''Vanilla''\right)\:$

$\mathrm{P2}≔\mathrm{DrawNormalSubgroupLattice}\left(G\,'\mathrm{indices}'\,'\mathrm{title}'=''Indices''\right)\:$

$\mathrm{P3}≔\mathrm{DrawNormalSubgroupLattice}\left(G\,'\mathrm{labels}'=''none''\,'\mathrm{title}'=''No\; Labels''\,'\mathrm{titlefont}'=\left[''Helvetica''\,10\right]\right)\:$

$\mathrm{P4}≔\mathrm{DrawNormalSubgroupLattice}\left(G\,'\mathrm{labels}'=''ids''\,'\mathrm{title}'=''Ids''\right)\:$

$\mathrm{P5}≔\mathrm{DrawNormalSubgroupLattice}\left(G\,'\mathrm{directed}'\,'\mathrm{title}'=''Directed''\right)\:$

$\mathrm{P6}≔\mathrm{DrawNormalSubgroupLattice}\left(G\,'\mathrm{highlight}'=\mathrm{Centre}\left(G\right)\,'\mathrm{title}'=''Centre\; Highlighted''\right)\:$

$\mathrm{plots}:-\mathrm{display}\left(\mathrm{Array}\left(\left[\left[\mathrm{P1}\,\mathrm{P2}\,\mathrm{P3}\right]\,\left[\mathrm{P4}\,\mathrm{P5}\,\mathrm{P6}\right]\right]\right)\right)$

$G≔\mathrm{DihedralGroup}\left(12\right)\:$

$\mathrm{plots}:-\mathrm{display}\left(\mathrm{Array}\left(\left[\mathrm{DrawSubgroupLattice}\left(G\,'\mathrm{labels}'='\mathrm{ids}'\,'\mathrm{title}'=''Full\; Subgroup\; Lattice''\right)\,\mathrm{DrawNormalSubgroupLattice}\left(G\,'\mathrm{labels}'='\mathrm{ids}'\,'\mathrm{title}'=''Normal\; Subgroup\; Lattice''\right)\right]\right)\right)$

$\mathrm{DrawNormalSubgroupLattice}\left(\mathrm{CayleyTableGroup}\left(\mathrm{DihedralGroup}\left(8\right)\right)\,'\mathrm{labels}'='\mathrm{ids}'\,'\mathrm{indices}'\right)$

$\mathrm{DrawNormalSubgroupLattice}\left(\mathrm{DihedralGroup}\left(6\,'\mathrm{form}'=''fpgroup''\right)\,'\mathrm{labels}'='\mathrm{ids}'\right)$

$G≔\mathrm{DirectProduct}\left(\mathrm{QuaternionGroup}\left(\right)\,\mathrm{CyclicGroup}\left(2\right)\right)$

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

$\mathrm{DrawNormalSubgroupLattice}\left(G\,'\mathrm{labels}'=''ids''\,'\mathrm{highlight}'=\left[\mathrm{Centre}\left(G\right)\,\mathrm{DerivedSubgroup}\left(G\right)\right]\right)$

$\mathrm{DrawNormalSubgroupLattice}\left(G\,'\mathrm{labels}'=''ids''\,'\mathrm{highlight}'=\left[\mathrm{Centre}\left(G\right)=''blue''\,\mathrm{DerivedSubgroup}\left(G\right)\right]\right)$

$G≔\mathrm{DihedralGroup}\left(6\right)$

$G≔{\mathbf{D}}_6$

$\mathrm{DrawNormalSubgroupLattice}\left(G\,'\mathrm{labels}'=''ids''\,'\mathrm{highlight}'=\left[\left[\mathrm{Centre}\left(G\right)\,\mathrm{DerivedSubgroup}\left(G\right)\right]=''blue''\,\mathrm{FittingSubgroup}\left(G\right)\right]\right)$

$G≔\mathrm{Group}\left(\left[\mathrm{Perm}\left(\left[\left[2\,9\right]\,\left[3\,15\right]\,\left[4\,5\right]\,\left[6\,7\right]\,\left[8\,28\right]\,\left[10\,12\right]\,\left[11\,32\right]\,\left[13\,14\right]\,\left[16\,21\right]\,\left[17\,23\right]\,\left[18\,22\right]\,\left[19\,31\right]\,\left[20\,30\right]\,\left[25\,27\right]\,\left[26\,44\right]\,\left[29\,48\right]\,\left[33\,38\right]\,\left[34\,40\right]\,\left[35\,39\right]\,\left[36\,47\right]\,\left[37\,46\right]\,\left[41\,43\right]\,\left[42\,60\right]\,\left[45\,62\right]\,\left[49\,54\right]\,\left[50\,56\right]\,\left[51\,55\right]\,\left[52\,61\right]\,\left[53\,58\right]\,\left[57\,59\right]\,\left[63\,64\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[1\,60\,7\,62\,24\,61\,6\,58\right]\,\left[2\,49\,10\,63\,13\,64\,3\,54\right]\,\left[4\,45\,18\,42\,27\,53\,17\,52\right]\,\left[5\,47\,23\,46\,25\,44\,22\,48\right]\,\left[8\,59\,20\,50\,19\,33\,11\,51\right]\,\left[9\,55\,15\,38\,14\,56\,12\,57\right]\,\left[16\,37\,35\,36\,43\,29\,34\,26\right]\,\left[21\,28\,40\,32\,41\,31\,39\,30\right]\right]\right)\right]\right)\:$

$\mathrm{DrawNormalSubgroupLattice}\left(G\,'\mathrm{highlight}'=\left[\mathrm{LowerCentralSeries}\left(G\right)=''blue''\,\mathrm{UpperCentralSeries}\left(G\right)=''red''\right]\right)$

$G≔\mathrm{SmallGroup}\left(256\,510\right)\:$

$\mathrm{DrawNormalSubgroupLattice}\left(G\,'\mathrm{labels}'=''ids''\,'\mathrm{highlight}'=\left[\mathrm{LowerCentralSeries}\left(G\right)=''blue''\,\mathrm{DerivedSeries}\left(G\right)=''red''\right]\right)$

$\mathrm{plots}:-\mathrm{display}\left(\left[\mathrm{seq}\right]\left(\mathrm{DrawNormalSubgroupLattice}\left(G\,'\mathrm{highlight}'=H\right)\,H=\mathrm{LowerCentralSeries}\left(G\right)\right)\,'\mathrm{insequence}'=\mathrm{true}\right)$

$G≔\mathrm{Group}\left(\left[\mathrm{Perm}\left(\left[\left[2\,9\right]\,\left[3\,15\right]\,\left[4\,5\right]\,\left[6\,7\right]\,\left[8\,28\right]\,\left[10\,12\right]\,\left[11\,32\right]\,\left[13\,14\right]\,\left[16\,21\right]\,\left[17\,23\right]\,\left[18\,22\right]\,\left[19\,31\right]\,\left[20\,30\right]\,\left[25\,27\right]\,\left[26\,44\right]\,\left[29\,48\right]\,\left[33\,38\right]\,\left[34\,40\right]\,\left[35\,39\right]\,\left[36\,47\right]\,\left[37\,46\right]\,\left[41\,43\right]\,\left[42\,60\right]\,\left[45\,62\right]\,\left[49\,54\right]\,\left[50\,56\right]\,\left[51\,55\right]\,\left[52\,61\right]\,\left[53\,58\right]\,\left[57\,59\right]\,\left[63\,64\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[1\,60\,7\,62\,24\,61\,6\,58\right]\,\left[2\,49\,10\,63\,13\,64\,3\,54\right]\,\left[4\,45\,18\,42\,27\,53\,17\,52\right]\,\left[5\,47\,23\,46\,25\,44\,22\,48\right]\,\left[8\,59\,20\,50\,19\,33\,11\,51\right]\,\left[9\,55\,15\,38\,14\,56\,12\,57\right]\,\left[16\,37\,35\,36\,43\,29\,34\,26\right]\,\left[21\,28\,40\,32\,41\,31\,39\,30\right]\right]\right)\right]\right)$

$G≔⟨\left(2\,9\right)\left(3\,15\right)\left(4\,5\right)\left(6\,7\right)\left(8\,28\right)\left(10\,12\right)\left(11\,32\right)\left(13\,14\right)\left(16\,21\right)\left(17\,23\right)\left(18\,22\right)\left(19\,31\right)\left(20\,30\right)\left(25\,27\right)\left(26\,44\right)\left(29\,48\right)\left(33\,38\right)\left(34\,40\right)\left(35\,39\right)\left(36\,47\right)\left(37\,46\right)\left(41\,43\right)\left(42\,60\right)\left(45\,62\right)\left(49\,54\right)\left(50\,56\right)\left(51\,55\right)\left(52\,61\right)\left(53\,58\right)\left(57\,59\right)\left(63\,64\right)\,\left(1\,60\,7\,62\,24\,61\,6\,58\right)\left(2\,49\,10\,63\,13\,64\,3\,54\right)\left(4\,45\,18\,42\,27\,53\,17\,52\right)\left(5\,47\,23\,46\,25\,44\,22\,48\right)\left(8\,59\,20\,50\,19\,33\,11\,51\right)\left(9\,55\,15\,38\,14\,56\,12\,57\right)\left(16\,37\,35\,36\,43\,29\,34\,26\right)\left(21\,28\,40\,32\,41\,31\,39\,30\right)⟩$

$\mathrm{lcs}≔\mathrm{LowerCentralSeries}\left(G\right)\:$

$\mathrm{ucs}≔\mathrm{UpperCentralSeries}\left(G\right)\:$

$\mathrm{plots}:-\mathrm{display}\left(\left[\mathrm{seq}\right]\left(\mathrm{DrawNormalSubgroupLattice}\left(G\,'\mathrm{labels}'=''ids''\,'\mathrm{highlight}'=\left[\mathrm{lcs}\left[i\right]=''ForestGreen''\,\mathrm{ucs}\left[i\right]=''Magenta''\right]\,'\mathrm{edgecolor}'=''pink''\,'\mathrm{title}'=''The\; Upper\; and\; Lower\; Central\; Series''\,'\mathrm{titlefont}'=\left[''Helvetica''\,''boldoblique''\,12\right]\right)\,i=1..\mathrm{numelems}\left(\mathrm{lcs}\right)\right)\,'\mathrm{insequence}'=\mathrm{true}\right)$