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A137315 a(n) is the least number m such that any finite group of order at least m has at least n automorphisms. 3


%S 1,3,7,7,13,13,19,19,31,31,31,31,43,43,43,43,61,61,61,61,67,67,67,67,

%T 91,91,91,91,91,91,91,91,121,121,121,121,127,127,127,127,151,151,151,

%U 151,151,151,151,151

%N a(n) is the least number m such that any finite group of order at least m has at least n automorphisms.

%C a(n) <= (n-1)^(n + (n-2)[log_2(n-1)]) for n > 4 [Ledermann, Neumann, Thm. 6.6].

%C a(n) is odd [MacHale, Sheehy, Thm. 15].

%C a(2n-1) = a(2n) for 1 < n < 204 [ibid.].

%C The case of cyclic groups shows that a(n)>=A139795(n). This inequality can be strict: if M denotes the Mathieu group M_{22} of order 2^7.3^, then Aut(12.M) = M.2, so that a(2^8.3^ + 1) > 2^9.3^, but A139795(2^8.3^ + 1) =^ + 1 < 2^9.3^

%H John N. Bray and Robert A. Wilson, <a href="https://doi.org/10.1112/S002460930400400X">On the orders of automorphism groups of finite groups</a>, Bull. London Math. Soc. 37 (2005) 381--385.

%H W. Ledermann, B. H. Neumann, <a href="https://doi.org/10.1098/rspa.1956.0006">On the order of the automorphism group of a finite group. I</a>, Proc. Roy. Soc. Lon., 233A(1195) (1956), 494-506

%H D. MacHale and R. Sheehy, <a href="http://www.jstor.org/stable/40656888">Finite groups with few automorphisms</a>, Math. Proc. Roy. Irish Acad., 104A(2) (2004), 231--238.

%H Benjamin Sambale, <a href="https://arxiv.org/abs/1909.13220">On a theorem of Ledermann and Neumann</a>, arXiv:1909.13220 [math.GR], 2019.

%e a(3) = a(4) = 7 because every finite group with at least 7 elements has at least 4 automorphisms while the cyclic group of order 6 has only phi(6)=2 automorphisms.

%Y Different from A139795 (see Comments).

%Y See also A340521.

%K nonn,hard,more

%O 1,2

%A _Benoit Jubin_, Apr 06 2008, May 26 2008

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