7.2. Alternating groups

The alternating group A(n>3), of order n!/2, can be generated using two permutaions of degree n. They are always

In[191]:=

AltGS[n_] := GenSet[Cycles[{1, 2, 3}], Cycles[Range[2, n]]]

The largest alternating group explicitly described in the Atlas is A(14):

In[192]:=

myTiming[A14SGS = SchreierSims[{}, AltGS[14], 14] ;]

0.013386 Second

In[193]:=

Length/@A14SGS

Out[193]=

StrongGenSet[12, 22]

In[194]:=

OrderOfGroup[A14SGS]

Out[194]=

43589145600

With xPerm` we can manipulate within seconds groups which are much larger

In[195]:=

myTiming[A30SGS = SchreierSims[{}, AltGS[30], 30] ;]

0.922208 Second

In[196]:=

Length/@A30SGS

Out[196]=

StrongGenSet[28, 54]

In[197]:=

OrderOfGroup[A30SGS]

Out[197]=

132626429906095529318154240000000

Going further takes almost a minute:

In[198]:=

myTiming[A50SGS = SchreierSims[{}, AltGS[50], 50] ;]

31.952438 Second

In[199]:=

Length/@A50SGS

Out[199]=

StrongGenSet[48, 94]

In[200]:=

OrderOfGroup[A50SGS]

Out[200]=

15207046600856689021806304083032384422188820784480256000000000000

In[201]:=

%//N

Out[201]=

1.5207*10^64


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