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- Characterization of finite $p$-groups by the order of their Schur multipliers ($t(G)=7$)
- Examples and properties
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For instance, the Schur multiplier of the nonabelian group of order 6 is the trivial group since every Sylow subgroup is cyclic. The Schur multiplier of the elementary abelian group of order 16 is an elementary abelian group of order 64, showing that the multiplier can be strictly larger than the group itself. The Schur multiplier of the quaternion group is trivial, but the Schur multiplier of dihedral 2-groups has order 2.
The Schur multipliers of the finite simple groups are given at the list of finite simple groups. The covering groups of the alternating and symmetric groups are of considerable recent interest. A projective representation is much like a group representation except that instead of a homomorphism into the general linear group GL n , C , one takes a homomorphism into the projective general linear group PGL n , C.
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In other words, a projective representation is a representation modulo the center. The Schur cover is also known as a covering group or Darstellungsgruppe. The Schur covers of the finite simple groups are known, and each is an example of a quasisimple group. The Schur cover of a perfect group is uniquely determined up to isomorphism, but the Schur cover of a general finite group is only determined up to isoclinism.
The study of such covering groups led naturally to the study of central and stem extensions. A central extension of a group G is an extension. If the group G is finite and one considers only stem extensions, then there is a largest size for such a group C , and for every C of that size the subgroup K is isomorphic to the Schur multiplier of G. If the finite group G is moreover perfect , then C is unique up to isomorphism and is itself perfect. Such C are often called universal perfect central extensions of G , or covering group as it is a discrete analog of the universal covering space in topology.
If the finite group G is not perfect, then its Schur covering groups all such C of maximal order are only isoclinic. It is also called more briefly a universal central extension , but note that there is no largest central extension, as the direct product of G and an abelian group form a central extension of G of arbitrary size.
Stem extensions have the nice property that any lift of a generating set of G is a generating set of C. These are all slightly later results of Schur, who also gave a number of useful criteria for calculating them more explicitly. In combinatorial group theory , a group often originates from a presentation. One important theme in this area of mathematics is to study presentations with as few relations as possible, such as one relator groups like Baumslag-Solitar groups.
These groups are infinite groups with two generators and one relation, and an old result of Schreier shows that in any presentation with more generators than relations, the resulting group is infinite.
Characterization of finite $p$-groups by the order of their Schur multipliers ($t(G)=7$)
The borderline case is thus quite interesting: finite groups with the same number of generators as relations are said to have a deficiency zero. For a group to have deficiency zero, the group must have a trivial Schur multiplier because the minimum number of generators of the Schur multiplier is always less than or equal to the difference between the number of relations and the number of generators, which is the negative deficiency.
An efficient group is one where the Schur multiplier requires this number of generators. A fairly recent topic of research is to find efficient presentations for all finite simple groups with trivial Schur multipliers. Jafari, S. Bulletin of the Iranian Mathematical Society , 43 7 , Bulletin of the Iranian Mathematical Society , 43, 7, , Bulletin of the Iranian Mathematical Society , ; 43 7 : Berkovich, On the order of the commutator subgroups and the Schur multiplier of a finite p -group, J.
Algebra Brown, D. Johnson and E.
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Examples and properties
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