P-core-free group

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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 G be a finite group and p be a prime number. We say that G is p-core-free if it satisfies the following equivalent conditions:

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.