A318895 Number of isoclinism classes of the groups of order 2^n.

1, 1, 1, 2, 3, 8, 27, 115

Offset: 0

Author

Jack W Grahl, Sep 05 2018

Keywords

• nonn
• more

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.

Examples

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

Links

• P. Hall, The classification of prime-power groups, J. Reine Angew. Math. 182 (1940), 130-141.
• Rodney James, M. F. Newman and E. A. O'Brien, The groups of order 128, Journal of Algebra, Volume 129, Issue 1 (1990), 136-158.
• Vipul Naik, This sequence, along with other properties of groups of order 2^n

Crossrefs

Cf. A000001, A000679. A000041 has an interpretation as the number of Abelian groups with order 2^n.