A235388 Number of groups of order 2n generated by involutions.

1, 1, 1, 2, 1, 1, 1, 4, 2, 1, 1, 4, 1, 1, 1, 12, 1, 3, 1, 3, 1, 1, 1, 11, 2, 1, 4, 3, 1, 3, 1, 49, 1, 1, 1, 12, 1, 1, 1, 9, 1, 2, 1, 3, 2, 1, 1, 46, 2, 3, 1, 3, 1, 8, 1, 9, 1, 1, 1, 10, 1, 1, 2, 359, 1, 2, 1, 3, 1, 2, 1, 40, 1, 1, 3, 3, 1, 2, 1, 38, 11, 1, 1

Offset: 1

Author

Eric M. Schmidt, Jan 08 2014

Keywords

• nonn

Comments

a(n) >= A104404(n). This can be proved using the characterization in A104404. Given an Abelian group G, the semidirect product G : , where h^2 = 1 and hgh = g^(-1) for any g in G, is generated by involutions. There is also a semidirect product Q8 : C2 generated by involutions. So an involution-generated group G : C2 exists for any finite group G that has all subgroups normal, and it can be shown that they are all nonisomorphic.

Links

• Eric M. Schmidt, Table of n, a(n) for n = 1..511

Programs

GAP

IsInvolutionGenerated := G -> Group(Filtered(G, g->g^2=Identity(G)))=G;
A235388 := function(n) local i, count; count := 0; for i in [1..NrSmallGroups(2*n)] do if IsInvolutionGenerated(SmallGroup(2*n, i)) then count := count + 1; fi; od; return count; end;