The SearchFrobeniusGroups( spec ) command searches Maple's database of Frobenius groups for groups satisfying properties specified in a sequence spec of search parameters. The valid search parameters may be grouped into several classes, as described in the following sections.

Use the output = X option to control the kind of output returned. By default, an expression sequence of results, corresponding to the output = "list" option is produced. Alternatively, you can request a simple count of the results by using the output = "count" option. In the event that the number of returned results is large, the output = "iterator" option is useful, as it returns an iterator object that you can use to iterate over the results one at a time.

Use the form = X option to control the form of the output from this command. By default, an expression sequence of IDs for the FrobeniusGroups database is returned. This is the same as specifying form = "id". To have an expression sequence of permutation groups, use the form = "permgroup" option. The form option has no meaning if the output = "count" option is specified.

Note that the IDs returned in the default case are the IDs of the groups within the FrobeniusGroups database.  These may differ from the IDs for the same group if it happens to be present in another database, such as the SmallGroups database, which has its own set of group IDs.

Note further that IDs returned by SearchFrobeniusGroups are limited to those actually present in the database. In particular, they are limited by the maximum group order and by the order exclusions documented in FrobeniusGroup.

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

$\mathrm{SearchFrobeniusGroups}\left('\mathrm{output}'=''count''\right)$

$9034$

$\mathrm{SearchFrobeniusGroups}\left('\mathrm{order}'=100\right)$

$\left[100\,1\right],\left[100\,2\right],\left[100\,3\right]$

$\mathrm{IsAbelian}\left(\mathrm{FrobeniusKernel}\left(\mathrm{FrobeniusGroup}\left(100\,1\right)\right)\right)$

$\mathrm{true}$

$\mathrm{IsAbelian}\left(\mathrm{FrobeniusKernel}\left(\mathrm{FrobeniusGroup}\left(100\,2\right)\right)\right)$

$\mathrm{true}$

$\mathrm{IsAbelian}\left(\mathrm{FrobeniusKernel}\left(\mathrm{FrobeniusGroup}\left(100\,3\right)\right)\right)$

$\mathrm{true}$

$\mathrm{IsCyclic}\left(\mathrm{FrobeniusKernel}\left(\mathrm{FrobeniusGroup}\left(100\,1\right)\right)\right)$

$\mathrm{true}$

$\mathrm{IsCyclic}\left(\mathrm{FrobeniusKernel}\left(\mathrm{FrobeniusGroup}\left(100\,2\right)\right)\right)$

$\mathrm{false}$

$\mathrm{IsCyclic}\left(\mathrm{FrobeniusKernel}\left(\mathrm{FrobeniusGroup}\left(100\,3\right)\right)\right)$

$\mathrm{false}$

$\mathrm{SearchFrobeniusGroups}\left(\mathrm{kernel}=25\,'\mathrm{cyclickernel}'\right)$

$\left[50\,1\right],\left[100\,1\right]$

$\mathrm{SearchFrobeniusGroups}\left(\mathrm{kernel}=25\,'\mathrm{cyclickernel}'\,'\mathrm{nilpotentcomplement}'\right)$

$\left[50\,1\right],\left[100\,1\right]$

$\mathrm{SearchFrobeniusGroups}\left('\mathrm{order}'\le 1000\,'\mathrm{complement}'=4\,'\mathrm{output}'=''count''\right)$

$59$

$\mathrm{SearchFrobeniusGroups}\left(10000<'\mathrm{order}'\&comma;1<'\mathrm{transitivity}'\&comma;'\mathrm{homocyclickernel}'\right)$

$\left[10100\&comma;1\right],\left[10506\&comma;1\right],\left[11342\&comma;1\right],\left[11772\&comma;1\right],\left[12656\&comma;1\right],\left[14520\&comma;1\right],\left[14520\&comma;2\right],\left[14520\&comma;3\right],\left[14520\&comma;4\right]$

$\mathrm{SearchFrobeniusGroups}\left(100\le '\mathrm{order}'\&comma;'\mathrm{order}'<300\&comma;'\mathrm{metabelian}'\&comma;'\mathrm{metacyclic}'=\mathrm{false}\right)$

$\left[100\&comma;2\right],\left[100\&comma;3\right],\left[126\&comma;2\right],\left[147\&comma;2\right],\left[147\&comma;3\right],\left[150\&comma;1\right],\left[150\&comma;3\right],\left[156\&comma;2\right],\left[162\&comma;2\right],\left[162\&comma;3\right],\left[162\&comma;4\right],\left[162\&comma;5\right],\left[180\&comma;1\right],\left[192\&comma;1\right],\left[192\&comma;2\right],\left[192\&comma;5\right],\left[196\&comma;1\right],\left[198\&comma;2\right],\left[200\&comma;1\right],\left[228\&comma;1\right],\left[234\&comma;2\right],\left[240\&comma;1\right],\left[242\&comma;2\right],\left[250\&comma;2\right],\left[250\&comma;3\right],\left[270\&comma;2\right],\left[270\&comma;3\right],\left[294\&comma;2\right],\left[294\&comma;3\right],\left[294\&comma;5\right]$

$\mathrm{SearchFrobeniusGroups}\left('\mathrm{rank}'=7\&comma;'\mathrm{output}'=''count''\right)$

$39$

$\mathrm{SearchFrobeniusGroups}\left('\mathrm{transitivity}'=1\&comma;'\mathrm{output}'=''count''\right)$

$8984$

$\mathrm{it}≔\mathrm{SearchFrobeniusGroups}\left('\mathrm{transitivity}'=1\&comma;'\mathrm{output}'=''iterator''\&comma;'\mathrm{form}'=''permgroup''\right)$

$\mathrm{it}≔\mathrm{\&lang;Iterator\; for\; FrobeniusGroups\; Query:\; "transitivity\; =\; 1",\; with\; 8984\; results\&rang;}$

$$\begin{gathered} \mathbf{for}\mathrm{id},G\mathbf{in}\mathrm{it}\mathbf{do} \\ \mathbf{if}\mathrm{IsPrimitive}\left(G\right)\mathbf{then} \\ \mathrm{print}\left(\mathrm{id}\right); \\ \mathbf{break} \\ \mathbf{end}\mathbf{if} \\ \mathbf{end}\mathbf{do}\&colon; \end{gathered}$$

$10,1$

The GroupTheory[SearchFrobeniusGroups] command was introduced in Maple 2019.

For more information on Maple 2019 changes, see Updates in Maple 2019.