

A318895


Number of isoclinism classes of the groups of order 2^n.


0




OFFSET

0,4


COMMENTS

The concept of isoclinism was introduced in Hall (1940) and is crucial to enumerating the groups of order p^n where p is a prime.
An isoclinism exists between two groups G1 and G2 if the following holds: There is an isomorphism f between their two inner automorphism groups G1/Z(G1) and G2/Z(G2). There is an isomorphism h between their two commutator groups [G1, G1] and [G2, G2]. Lastly, f and h commute with F1 and F2, where F1 is the mapping from G1/Z(G1) x G1/Z(G1) to [G1, G1], given by a, b > ab(a^1)(b^1), and F2 is defined analogously.


LINKS

Table of n, a(n) for n=0..7.
P. Hall, The classification of primepower groups, J. Reine Angew. Math. 182 (1940), 130141.
Rodney James, M. F. Newman and E. A. O'Brien, The groups of order 128, Journal of Algebra, Volume 129, Issue 1 (1990), 136158.
Vipul Naik, This sequence, along with other properties of groups of order 2^n


EXAMPLE

There are 51 groups of order 32. These fall into 8 isoclinism classes. Therefore a(5) = 8.


CROSSREFS

Cf. A000001, A000679. A000041 has an interpretation as the number of Abelian groups with order 2^n.
Sequence in context: A041503 A086613 A121401 * A093858 A080568 A091339
Adjacent sequences: A318892 A318893 A318894 * A318896 A318897 A318898


KEYWORD

nonn,more


AUTHOR

Jack W Grahl, Sep 05 2018


STATUS

approved



