Elementary group
In algebra, more specifically group theory, a p-elementary group is a direct product of a finite cyclic group of order relatively prime to p and a p-group. A finite group is an elementary group if it is p-elementary for some prime number p. An elementary group is nilpotent. Brauer's theorem on induced characters states that a character on a finite group is a linear combination with integer coefficients of characters induced from elementary subgroups.
More generally, a finite group G is called a p-hyperelementary if it has the extension
- [math]\displaystyle{ 1 \longrightarrow C \longrightarrow G \longrightarrow P \longrightarrow 1 }[/math]
where [math]\displaystyle{ C }[/math] is cyclic of order prime to p and P is a p-group. Not every hyperelementary group is elementary: for instance the non-abelian group of order 6 is 2-hyperelementary, but not 2-elementary.
See also
References
- Arthur Bartels, Wolfgang Lück, Induction Theorems and Isomorphism Conjectures for K- and L-Theory
- G. Segal, The representation-ring of a compact Lie group
- J.P. Serre, "Linear representations of finite groups". Graduate Texts in Mathematics, vol. 42, Springer-Verlag, New York, Heidelberg, Berlin, 1977,
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