In mathematics, in the realm of group theory, a group is said to be superperfect when its first two homology groups are trivial: H1(G, Z) = H2(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 and Schur multiplier both vanish; abelianization equals the first homology, while the Schur multiplier equals the second homology.
Other articles related to "superperfect group, group, superperfect":
... For example, if G is the fundamental group of a homology sphere, then G is superperfect ... The smallest finite, non-trivial superperfect group is the binary icosahedral group (the fundamental group of the Poincaré homology sphere) ... The alternating group A5 is perfect but not superperfect it has a non-trivial central extension, the binary icosahedral group (which is in fact its UCE, and is superperfect) ...
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