A137316 Array read by rows: T(n,k) is the number of automorphisms of the k-th group of order n, where the ordering is such that the rows are nondecreasing.
1, 1, 2, 2, 6, 4, 2, 6, 6, 4, 8, 8, 24, 168, 6, 48, 4, 20, 10, 4, 12, 12, 12, 24, 12, 6, 42, 8, 8, 16, 16, 16, 32, 32, 32, 32, 48, 64, 96, 192, 192, 20160, 16, 6, 12, 48, 54, 432, 18, 8, 20, 24, 40, 40, 12, 42, 10, 110, 22, 8, 16, 16, 24, 24, 24, 24, 24, 24, 48, 48, 48, 48, 144, 336
Offset: 1
Examples
The table begins as follows: 1 1 2 2 6 4 2 6 6 4 8 8 24 168 6 48 4 20 10 4 12 12 12 24 12 6 42 The first row with two numbers corresponds to the two groups of order 4, the cyclic group Z_4 and the Klein group Z_2 x Z_2, whose automorphism groups are respectively the group (Z_4)^* = Z_2 and the symmetric group S_3.
Links
- D. MacHale and R. Sheehy, Finite groups with few automorphisms, Math. Proc. Roy. Irish Acad., 104A(2) (2004), 231--238.
Programs
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GAP
# GAP 4 Print("\n") ; for o in [ 1 .. 33 ] do n := NumberSmallGroups(o) ; og := [] ; for i in [1 .. n] do g := SmallGroup(o,i) ; H := AutomorphismGroup(g) ; ho := Order(H) ; Add(og,ho) ; od; Sort(og) ; Print(og) ; Print("\n") ; od; # R. J. Mathar, Jul 13 2013
Comments