A340521 -id:A340521 - cp's OEIS frontend

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

1, 3, 7, 7, 13, 13, 19, 19, 31, 31, 31, 31, 43, 43, 43, 43, 61, 61, 61, 61, 67, 67, 67, 67, 91, 91, 91, 91, 91, 91, 91, 91, 121, 121, 121, 121, 127, 127, 127, 127, 151, 151, 151, 151, 151, 151, 151, 151

Offset: 1

Benoit Jubin, Apr 06 2008, May 26 2008

a(n) <= (n-1)^(n + (n-2)[log_2(n-1)]) for n > 4 [Ledermann, Neumann, Thm. 6.6].
a(n) is odd [MacHale, Sheehy, Thm. 15].
a(2n-1) = a(2n) for 1 < n < 204 [ibid.].
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 2^7.3^2.5.7.11, then Aut(12.M) = M.2, so that a(2^8.3^2.5.7.11 + 1) > 2^9.3^3.5.7.11, but A139795(2^8.3^2.5.7.11 + 1) = 2.3.5.7^2.11.13.23 + 1 < 2^9.3^3.5.7.11.

			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.

John N. Bray and Robert A. Wilson, On the orders of automorphism groups of finite groups, Bull. London Math. Soc. 37 (2005) 381--385.
W. Ledermann, B. H. Neumann, On the order of the automorphism group of a finite group. I, Proc. Roy. Soc. Lon., 233A(1195) (1956), 494-506
D. MacHale and R. Sheehy, Finite groups with few automorphisms, Math. Proc. Roy. Irish Acad., 104A(2) (2004), 231--238.
Benjamin Sambale, On a theorem of Ledermann and Neumann, arXiv:1909.13220 [math.GR], 2019.

Different from A139795 (see Comments).

A341823 Number of finite groups G with |Aut(G)| = 2^n.

Original entry on oeis.org

2, 3, 4, 7, 11, 19, 34, 70

Offset: 0

Des MacHale, Feb 20 2021

This sequence is infinite, but the amount of computation needed to consider the large number of groups of order 2^8 suggests it may be hard to find the next term.

			a(3) = 7, because there are seven finite groups G with |Aut(G)| = 8. Four cyclic groups: Aut(C_15) = Aut(C_16) = Aut(C_20) = Aut(C_30) ~~ C_4 x C_2, also Aut(C_4 x C_2) = Aut(D_4) ~~ D_4, with D_4 is the dihedral group of the square, finally Aut(C_24) ~~ C_2 x C_2 x C_2 = (C_2)^3 where ~~ stands for “isomorphic to". - _Bernard Schott_, Mar 04 2021

J. Flynn, D. MacHale, E. A. O'Brien and R. Sheehy, Finite Groups whose Automorphism Groups are 2-groups, Proc. Royal Irish Academy, 94A, (2) 1994, 137-145.

Cf. A341824, A341825.

Subsequence of A340521.

Offset modified by Bernard Schott, Mar 04 2021

A341824 Number of groups of order 2^n which occur as Aut(G) for some finite group G.

Original entry on oeis.org

1, 1, 2, 3, 4, 9, 14, 33

Offset: 0

Des MacHale, Feb 26 2021

The number of groups of order 2^n is A000679(n); the percentage of the 2-groups which occur as automorphism groups appears to decrease as n increases: 100, 100, 100, 60, 28.5, 17.6, 5.2, 1.4, ...

Jianing Song remarks that it is also interesting to consider infinite groups, and asks if there is an infinite group G with Aut(G) isomorphic to C_8. Des MacHale, Mar 03 2021, replies that at present this is not known. [Comment edited by N. J. A. Sloane, Mar 07 2021]

			a(5) = 9 because there are nine groups of order 32 which occur as automorphism groups of finite groups.
From _Bernard Schott_, Feb 26 2021: (Start)
Aut(C_15) = Aut(C_16) = Aut(C_20) = Aut(C_30) ~~ C_4 x C_2 where ~~ stands for "isomorphic to".
Aut(C_4 x C_2) = Aut(D_4) ~~ D_4 where D_4 is the dihedral group of the square.
Aut(C_24) ~~ C_2 x C_2 x C_2 = (C_2)^3.
There exist five groups of order 8 (A054397), the three groups C_4 x C_2, D_4, C_2 x C_2 x C_2 occur as automorphim groups of order 8, but the cyclic group C_8 and the quaternions group Q_8 never occur as Aut(G) for some finite G, so a(3) = 3. (End)

J. Flynn, D. MacHale, E. A. O'Brien and R. Sheehy, Finite Groups whose Automorphism Groups are 2-groups, Proc. Royal Irish Academy, 94A, (2) 1994, 137-145.

Cf. A000679, A054397, A340521, A341823, A341825.

a(n) <= A000679(n). - Des MacHale, Mar 02 2021

Offset modified by Jianing Song, Mar 02 2021

A341825 Number of finite groups G with |Aut(G)| = n.

Original entry on oeis.org

2, 3, 0, 4, 0, 6, 0, 7, 0, 2, 0, 9, 0, 0, 0, 11, 0, 4, 0, 7, 0, 2, 0, 22, 0, 0, 0, 2, 0, 2, 0, 19, 0, 0, 0, 12, 0, 0, 0, 14, 0, 7, 0, 3, 0, 2, 0

Offset: 1

Des MacHale, Mar 02 2021

The smallest odd index n > 1 for which a(n) > 0 is 2187 = 3^7 (see A340521).

There exist even indices n that are not values taken by totient function phi (A002202) for which a(n) > 0. For example, John Bray has produced a group G that is the semidirect product 19:9 of order 3^2*19 = 171 such that |Aut(G)| = 1026 = 2*3^3*19.

			a(6) = 6, because there are six groups G with |Aut(G)| = 6. Four  cyclic groups: Aut(C_7) = Aut(C_9) = Aut(C_14) = Aut(C_18) ~~ C_6, and also Aut(C_2 x C_2) = Aut(S_3) ~~ S_3, where ~~ stands for “isomorphic to”. - _Bernard Schott_, Mar 02 2021
a(8) = 7, because there are seven groups G with |Aut(G)| = 8.

D. MacHale and R. Sheehy, Finite groups with few automorphisms, Mathematical Proceedings of the Royal Irish Academy, Vol. 104A, No. 2 (December 2004), 231-238.

Cf. A002202, A137315, A340521, A341823, A341824.

a(2) = 3, a(p) = 0 if p odd prime.

a(A002202(n)) > 0, since |Aut(C_n)| = phi(n).

A368480 Orders k such that there exists a group of order k with automorphism group of odd order.

Original entry on oeis.org

1, 2, 2187

Offset: 1

Robin Jones, Dec 26 2023

			a(1)=1 since the trivial group has automorphism group of order 1. a(2)=2 since C_2 has automorphism group of order 1.

Peter Hegarty and Desmond MacHale, Minimal Odd Order Automorphism Groups, Journal of Group Theory, 13 (2009), 243-255; arXiv:0905.0993 [math.GR], 2009.

Cf. A340521, A365179.

cp's OEIS Frontend

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

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A341823 Number of finite groups G with |Aut(G)| = 2^n.

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A341824 Number of groups of order 2^n which occur as Aut(G) for some finite group G.

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A341825 Number of finite groups G with |Aut(G)| = n.

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A368480 Orders k such that there exists a group of order k with automorphism group of odd order.

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