IsAlternating - Maple Help
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GroupTheory

 IsAlternating
 Monte-Carlo test for alternating groups
 IsSymmetric
 Monte-Carlo test for symmetric groups

 Calling Sequence IsAlternating( G ) IsAlternating( G, confidence = val ) IsSymmetric( G ) IsSymmetric( G, confidence = val )

Parameters

 G - a permutation group val - confidence level; a number between 0 and 1

Description

 • 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.

Examples

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

Note these first examples are abstractly isomorphic to the indicated group, but are not permutation equivalent to it in its natural action.

 > $\mathrm{AreIsomorphic}\left(\mathrm{SmallGroup}\left(12,3\right),\mathrm{Alt}\left(4\right)\right)$
 ${\mathrm{true}}$ (1)
 > $\mathrm{IsAlternating}\left(\mathrm{SmallGroup}\left(12,3\right)\right)$
 ${\mathrm{false}}$ (2)

Construct the regular representation of the symmetric group of degree $3$.

 > $G≔\mathrm{Group}\left(\left[\mathrm{Perm}\left(\left[\left[1,2\right],\left[3,6\right],\left[4,5\right]\right]\right),\mathrm{Perm}\left(\left[\left[1,3,4\right],\left[2,5,6\right]\right]\right)\right]\right)$
 ${G}{≔}⟨\left({1}{,}{2}\right)\left({3}{,}{6}\right)\left({4}{,}{5}\right){,}\left({1}{,}{3}{,}{4}\right)\left({2}{,}{5}{,}{6}\right)⟩$ (3)
 > $\mathrm{AreIsomorphic}\left(G,\mathrm{Symm}\left(3\right)\right)$
 ${\mathrm{true}}$ (4)
 > $\mathrm{IsSymmetric}\left(G\right)$
 ${\mathrm{false}}$ (5)
 > $G≔\mathrm{Group}\left(\left[\mathrm{seq}\right]\left(\mathrm{Perm}\left(\left[\left[i,i+1,i+2\right]\right]\right),i=1..8\right)\right)$
 ${G}{≔}⟨\left({1}{,}{2}{,}{3}\right){,}\left({2}{,}{3}{,}{4}\right){,}\left({3}{,}{4}{,}{5}\right){,}\left({4}{,}{5}{,}{6}\right){,}\left({5}{,}{6}{,}{7}\right){,}\left({6}{,}{7}{,}{8}\right){,}\left({7}{,}{8}{,}{9}\right){,}\left({8}{,}{9}{,}{10}\right)⟩$ (6)
 > $\mathrm{IsAlternating}\left(G\right)$
 ${\mathrm{true}}$ (7)
 > $\mathrm{IsSymmetric}\left(\mathrm{Group}\left(\left[\mathrm{seq}\right]\left(\mathrm{Perm}\left(\left[\left[i,i+1\right]\right]\right),i=1..9\right)\right)\right)$
 ${\mathrm{true}}$ (8)

By decreasing the value of the confidence option to 1/2, we can virtually guarantee incorrect answers some of the time.

 > $\mathrm{seq}\left(\mathrm{IsAlternating}\left(G,':-\mathrm{confidence}'=\frac{1}{2}\right),i=1..10\right)$
 ${\mathrm{true}}{,}{\mathrm{true}}{,}{\mathrm{true}}{,}{\mathrm{true}}{,}{\mathrm{true}}{,}{\mathrm{true}}{,}{\mathrm{true}}{,}{\mathrm{true}}{,}{\mathrm{true}}{,}{\mathrm{true}}$ (9)

Compatibility

 • The GroupTheory[IsAlternating] and GroupTheory[IsSymmetric] commands were introduced in Maple 17.
 • For more information on Maple 17 changes, see Updates in Maple 17.