# Superperfect group

In

mathematics , in the realm ofgroup theory , a group is said to be**superperfect**when its first two homology groups are trivial.The first homology group of a group is the abelianization of the group itself, since the homology of a group "G" is the homology of any

Eilenberg-MacLane space of type "K"("G",1); thefundamental group 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 perfect group is superperfect if and only if it is its own universal central extension.For example, if "G" is the fundamental group of a

homology sphere , then "G" is superperfect. The smallest finite, non-trivial superperfect group is thebinary icosahedral group (the fundamental group of the Poincaré homology sphere).Every acyclic group 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. MathSciNet|id=2009444

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Perfect group**— In mathematics, in the realm of group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no nontrivial abelian quotients.The smallest (non trivial) perfect group is the alternating… … Wikipedia**Schur multiplier**— In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group of a group G. It was introduced by Issai Schur (1904) in his work on projective representations. Contents 1 Examples and properties 2 Re … Wikipedia**List of mathematics articles (S)**— NOTOC S S duality S matrix S plane S transform S unit S.O.S. Mathematics SA subgroup Saccheri quadrilateral Sacks spiral Sacred geometry Saddle node bifurcation Saddle point Saddle surface Sadleirian Professor of Pure Mathematics Safe prime Safe… … Wikipedia**Acyclic space**— In mathematics, an acyclic space is a topological space X in which cycles are always boundaries, in the sense of homology theory. This implies that the integral homology groups in all dimensions of X are isomorphic to the corresponding homology… … Wikipedia**Covering groups of the alternating and symmetric groups**— In the mathematical area of group theory, the covering groups of the alternating and symmetric groups are groups that are used to understand the projective representations of the alternating and symmetric groups. The covering groups were… … Wikipedia**Semi-s-cobordism**— In mathematics, a cobordism ( W , M , M −) of an ( n + 1) dimensionsal manifold (with boundary) W between its boundary components, two n manifolds M and M − (n.b.: the original creator of this topic, Jean Claude Hausmann, used the notation M −… … Wikipedia**Prime number**— Prime redirects here. For other uses, see Prime (disambiguation). A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is… … Wikipedia**Divisor**— divisible redirects here. For divisibility of groups, see Divisible group. For the second operand of a division, see Division (mathematics). For divisors in algebraic geometry, see Divisor (algebraic geometry). For divisibility in the ring theory … Wikipedia**100000 (number)**— List of numbers – Integers 10000 100000 1000000 Cardinal One hundred thousand Ordinal One hundred thousandth Factorization 25 · 55 Roman numeral C Roman numeral (Unicode) … Wikipedia