P-core-free group
From Groupprops
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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Definition
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.