The article defines a property of groups, where the definition may be in terms of a particular prime that serves as parameter
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Let be a finite group and be a prime number. We say that is -core-free if it satisfies the following equivalent conditions:
- The -core (i.e., the largest normal -subgroup of ) is trivial. The -core is also termed the Sylow-core, and is the normal core of any Sylow subgroup. It also equals the pi-core where .
- has no nontrivial normal -subgroup.
- possesses a faithful irreducible representation over a field of characteristic .
Equivalence of definitions
The first two definitions are clearly equivalent. For equivalence with the third definition, refer p-core-free iff there exists a faithful irreducible representation over a field of characteristic p.