Isoclinism of groups

In mathematics, specifically group theory, isoclinism is an equivalence relation on groups which generalizes isomorphism. Isoclinism was introduced by Hall (1940) to help classify and understand p-groups, although it is applicable to all groups. Isoclinism also has consequences for the Schur multiplier and the associated aspects of character theory, as described in Suzuki (1982, p. 256) and Conway et al. (1985, p. xxiii, Ch. 6.7). The word "isoclinism" comes from the Greek ισοκλινης meaning equal slope.

Some textbooks discussing isoclinism include Berkovich (2008, §29) and Blackburn, Neumann & Venkataraman (2007, §21.2) and Suzuki (1986, pp. 92–95).

Definition

The isoclinism class of a group G is determined by the groups G/Z(G) (the inner automorphism group) and G′ (the commutator subgroup) and the commutator map from G/Z(G) × G/Z(G) to G′ (taking a, b to aba⁻¹b⁻¹).

In other words, two groups G₁ and G₂ are isoclinic if there are isomorphisms from G₁/Z(G₁) to G₂/Z(G₂) and from G₁′ to G₂′ commuting with the commutator map.

Examples

All Abelian groups are isoclinic since they are equal to their centers and their commutator subgroups are always the identity subgroup. Indeed, a group is isoclinic to an abelian group if and only if it is itself abelian, and G is isoclinic with G×A if and only if A is abelian. The dihedral, quasidihedral, and quaternion groups of order 2ⁿ are isoclinic for n≥3, Berkovich (2008, p. 285) in more detail.

Isoclinism divides p-groups into families, and the smallest members of each family are called stem groups. A group is a stem group if and only if Z(G) ≤ [G,G], that is, if and only if every element of the center of the group is contained in the derived subgroup (also called the commutator subgroup), Berkovich (2008, p. 287). Some enumeration results on isoclinism families are given in Blackburn, Neumann & Venkataraman (2007, p. 226).

Isoclinism is used in theory of projective representations of finite groups, as all Schur covering groups of a group are isoclinic, a fact already hinted at by Hall according to Suzuki (1982, p. 256). This is used in describing the character tables of the finite simple groups (Conway et al. 1985, p. xxiii, Ch. 6.7).

References

• Berkovich, Yakov (2008), Groups of prime power order. Vol. 1, de Gruyter Expositions in Mathematics, vol. 46, Walter de Gruyter GmbH & Co. KG, Berlin, doi:10.1515/9783110208221.285, ISBN 978-3-11-020418-6, MR 2464640
• Blackburn, Simon R.; Neumann, Peter M.; Venkataraman, Geetha (2007), Enumeration of finite groups, Cambridge Tracts in Mathematics no 173 (1st ed.), Cambridge University Press, ISBN 978-0-521-88217-0, OCLC 154682311
• Conway, John Horton; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A. (1985), Atlas of finite groups, Oxford University Press, ISBN 978-0-19-853199-9, MR 0827219
• Hall, Philip (1940), "The classification of prime-power groups", Journal für die reine und angewandte Mathematik, 1940 (182): 130–141, doi:10.1515/crll.1940.182.130, ISSN 0075-4102, MR 0003389, S2CID 122817195
• Struik, Ruth Rebekka (1960). "A note on prime-power groups". Canadian Mathematical Bulletin. 3: 27–30. doi:10.4153/cmb-1960-006-5. ISSN 0008-4395. MR 0148744.
• Suzuki, Michio (1982), Group theory. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 247, Berlin, New York: Springer-Verlag, ISBN 978-3-540-10915-0, MR 0648772
• Suzuki, Michio (1986), Group theory. II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 248, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-642-86885-6, ISBN 978-0-387-10916-9, MR 0815926