mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, in the realm of

group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...

, a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

is said to be superperfect when its first two

homology group In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...

s are

trivial Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense. Latin Etymology The ancient Romans used the word ''triviae'' to describe where one road split or forked ...

: ''H''₁(''G'', Z) = ''H''₂(''G'', Z) = 0. This is stronger than a perfect group, which is one whose first homology group vanishes. In more classical terms, a superperfect group is one whose

abelianization In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important because it is the smallest normal s ...

and

Schur multiplier In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2(G, \Z) of a group ''G''. It was introduced by in his work on projective representations. Examples and properties The Schur multiplier \oper ...

both vanish; abelianization equals the first homology, while the Schur multiplier equals the second homology.

Definition

The first homology group of a group is the

of the group itself, since the homology of a group ''G'' is the homology of any

Eilenberg–MacLane space In mathematics, specifically algebraic topology, an Eilenberg–MacLane spaceSaunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name. ...

of type ''K''(''G'', 1); the

fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...

of a ''K''(''G'', 1) is ''G'', and the first homology of ''K''(''G'', 1) is then abelianization of its fundamental group. Thus, if a group is superperfect, then it is perfect. A

finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...

perfect group is superperfect if and only if it is its own

universal central extension In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2(G, \Z) of a group ''G''. It was introduced by in his work on projective representations. Examples and properties The Schur multiplier \oper ...

(UCE), as the second homology group of a perfect group parametrizes central extensions.

Examples

For example, if ''G'' is the fundamental group of a

homology sphere Homology may refer to: Sciences Biology *Homology (biology), any characteristic of biological organisms that is derived from a common ancestor *Sequence homology, biological homology between DNA, RNA, or protein sequences *Homologous chromo ...

, then ''G'' is superperfect. The smallest finite, non-trivial superperfect group is the

binary icosahedral group In mathematics, the binary icosahedral group 2''I'' or Coxeter&Moser: Generators and Relations for discrete groups: : Rl = Sm = Tn = RST is a certain nonabelian group of order 120. It is an extension of the icosahedral group ''I'' or (2,3,5) of o ...

(the fundamental group of the Poincaré homology sphere). The alternating group ''A''₅ is perfect but not superperfect: it has a non-trivial central extension, the binary icosahedral group (which is in fact its UCE) is superperfect. More generally, the

projective special linear group In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associate ...

s PSL(''n'', ''q'') are simple (hence perfect) except for PSL(2, 2) and PSL(2, 3), but not superperfect, with the

special linear group In mathematics, the special linear group of degree ''n'' over a field ''F'' is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the genera ...

s SL(''n'',''q'') as central extensions. This family includes the binary icosahedral group (thought of as SL(2, 5)) as UCE of ''A''₅ (thought of as PSL(2, 5)). Every

acyclic group In mathematics, an acyclic space is a nonempty topological space ''X'' in which cycles are always boundaries, in the sense of homology theory. This implies that integral homology groups in all dimensions of ''X'' are isomorphic to the correspond ...

is superperfect, but the converse is not true: the binary icosahedral group is superperfect, but not acyclic.

References

* A. Jon Berrick and Jonathan A. Hillman, "Perfect and acyclic subgroups of finitely presentable groups", ''Journal of the London Mathematical Society'' (2) 68 (2003), no. 3, 683--698. Properties of groups {{Abstract-algebra-stub