Superperfect Group
Group Theory, Group Cohomology, Schur Multiplier,Commutator Subgroup, Eilenberg MacLane Space
Herausgegeben: Waldorm, Elwood Kuni
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, in the realm of group theory, a group is said to be superperfect when its first two homology groups are trivial: H_1(G;mathbf{Z})=H_2(G;mathbf{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.The first hom...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, in the realm of group theory, a group is said to be superperfect when its first two homology groups are trivial: H_1(G;mathbf{Z})=H_2(G;mathbf{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.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); the fundamental 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.
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