The commands IsAlternating( G ) and IsSymmetric( G ) provide one-sided Monte-Carlo tests for a permutation group G to be, respectively, an alternating or symmetric group in its natural action on the set $\left\{1\,2\,\dots \,n\right\}$, where n is the degree of G.

If the command returns the value true, then the result is guaranteed to be correct.  However, it may return the value false incorrectly, with small probability.

The level of confidence can be controlled by means of the confidence option. By default, the confidence level is set to 999999/1000000, which is the likelihood that either command IsAlternating or IsSymmetric returns the value false when the input group is actually a symmetric or alternating group, respectively. A higher value of the confidence option requires an increase in running time. Likewise, setting the confidence option to a lower value reduces the running time, but increases the chance that an incorrect false value is returned.

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

$\mathrm{AreIsomorphic}\left(\mathrm{SmallGroup}\left(12\,3\right)\,\mathrm{Alt}\left(4\right)\right)$

$\mathrm{true}$

$\mathrm{IsAlternating}\left(\mathrm{SmallGroup}\left(12\,3\right)\right)$

$\mathrm{false}$

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

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

$\mathrm{AreIsomorphic}\left(G\,\mathrm{Symm}\left(3\right)\right)$

$\mathrm{true}$

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

$\mathrm{false}$

$G≔\mathrm{Group}\left(\left[\mathrm{seq}\right]\left(\mathrm{Perm}\left(\left[\left[i\,i+1\,i+2\right]\right]\right)\,i=1..8\right)\right)$

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

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

$\mathrm{true}$

$\mathrm{IsSymmetric}\left(\mathrm{Group}\left(\left[\mathrm{seq}\right]\left(\mathrm{Perm}\left(\left[\left[i\,i+1\right]\right]\right)\,i=1..9\right)\right)\right)$

$\mathrm{true}$

$\mathrm{seq}\left(\mathrm{IsAlternating}\left(G\,':-\mathrm{confidence}'=\frac{1}{2}\right)\,i=1..10\right)$

$\mathrm{true},\mathrm{true},\mathrm{true},\mathrm{true},\mathrm{true},\mathrm{true},\mathrm{true},\mathrm{true},\mathrm{true},\mathrm{true}$

The GroupTheory[IsAlternating] and GroupTheory[IsSymmetric] commands were introduced in Maple 17.

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