A139669 - OEIS

A139669

Number of isomorphism classes of finite groups of order 11*2^n.

1, 2, 4, 12, 42, 195, 1387, 19324, 1083472

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OFFSET

0,2

COMMENTS

This appears to be the smallest possible number of groups of order q*2^n for an odd number q.

Apparently, a(n) is also the number of isomorphism classes of finite groups of order 19*2^n and, more generally, of order p*2^n for primes p such that p is congruent to 3 modulo 4 and p+1 is not a power of 2.

Comment from Miles Englezou, Sep 26 2024: (Start)

The comment above which starts "Apparently, ... power of 2." is not true. (For a proof see the Miles Englezou link). However, it is true that a(0) to a(8) are the smallest possible number of groups of order q*2^n for an odd number q, and this can be generalized in the way stated below. (For further details see the Miles Englezou link).

A correct generalization of the 9 terms:

The number of groups of order q*2^n is the least possible for prime q such that q == 3 (mod 4) and where the least 2^m such that 2^m == 1 (mod q) is greater than 2^n. Or put another way, if A014664(A080148(n)) > n, then for q = A000040(A080148(n)) the number of groups of order q*2^n is the least possible. (End)

REFERENCES

J. H. Conway et al., The Symmetries of Things, Peters, 2008, p. 206.

LINKS

Table of n, a(n) for n=0..8.

John H. Conway, Heiko Dietrich and E. A. O'Brien, Counting groups: gnus, moas and other exotica.

Miles Englezou, Proofs

FORMULA

a(n) = A000001(11*2^n). - Max Alekseyev, Apr 26 2010

EXAMPLE

a(2) is the number of groups of order 11*2^2=44, which is 4 and also the number of groups of order 19*2^2=76, 23*2^2=92, etc.

MAPLE

A139669 := n -> GroupTheory[NumGroups](11*2^n);