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A235388 Number of groups of order 2n generated by involutions. 3

%I

%S 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,

%T 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,

%U 2,1,3,1,2,1,40,1,1,3,3,1,2,1,38,11,1,1

%N Number of groups of order 2n generated by involutions.

%C a(n) >= A104404(n). This can be proved using the characterization in A104404. Given an Abelian group G, the semidirect product G : <h>, 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.

%H Eric M. Schmidt, <a href="/A235388/b235388.txt">Table of n, a(n) for n = 1..511</a>

%o (GAP)

%o IsInvolutionGenerated := G -> Group(Filtered(G, g->g^2=Identity(G)))=G;

%o 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;

%K nonn

%O 1,4

%A _Eric M. Schmidt_, Jan 08 2014

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Last modified January 24 18:53 EST 2022. Contains 350565 sequences. (Running on oeis4.)