A341824 -id:A341824 - cp's OEIS frontend

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

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

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).

cp's OEIS Frontend

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

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