A (2,*)-group is a triple (G,x,g), where G is a finite group and x,g are elements of G
such that G= and x is an involution. Two (2,*)-groups (G1,x1,g1) and (G2,x2,g2)
are isomorphic iff there is a group isomorphism f:G1->G2 with f(x1)=x2, f(g1)=g2.
A group G for which there is a (2,*)-group (G1,x1,g1) with G isomorphic to G1 is
said to be (2,*)-generated.
All (2,*)-groups of order at most 6000 are now known. There are 129340 (2,*)-generated
groups, giving rise to 345070 (2,*)-groups of order at most 6000.
The file TwoStarGroups.mgm contains two functions, TwoStarGroups(MaxOrder) and
TwoStarPairs(MaxOrder).
The function TwoStarGroups(MaxOrder) returns a double sequence, say L, where,
for an integer m, 1<=m<=min(6000,MaxOrder), the sequence L[m] contains all
(2,*)-generated groups of order m. In fact, L[m,i] is a record where L[m,i]`grp is
the i-th (2,*)-generated group of order m and L[m,i]`type is either "PC" or "Perm",
depending on whether the group is soluble (in which case it's presented as a PC-group)
or insoluble (in which case it's presented as a permutation group).
The function requires the file "TwoStarGroups6k.txt".
Construction of the sequence with MaxOrd=6000 takes about 1 minute on a MacBook.
The function TwoStarPairs returns a triple sequence, say P, of all (2,*)-groups of order
at most min(MaxOrd,6000). Here P[m,i,k] is the permutation group G of order m, generated
by involution G.1 and another elements G.2. Each (2,*)-group (G,x,g) of order at most
min(MaxOrd,6000) appears in this list precisely once as (G,G.1,G.2).
Moreover, the group P[m,i,k] is isomorphic to the group L[m,i], where L is the output
of the function TwoStarGroups(MaxOrd).
The function requires the file "TwoStarPairs6k.txt".
Construction of the sequence with MaxOrd=6000 may take a very long time if run on a
laptop, and close to 10 minutes on a decent server.