P-normal not implies p-solvable

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This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., p-normal group) need not satisfy the second subgroup property (i.e., p-solvable group)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications
Get more facts about p-normal group|Get more facts about p-solvable group
EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property p-normal group but not p-solvable group|View examples of subgroups satisfying property p-normal group and p-solvable group

Statement

We can have a finite group G and a prime number p such that G is a p-normal group but is not a p-solvable group.

Facts used

  1. A-group implies p-normal for all p

Proof

Example of the alternating group of degree five

Further information: alternating group:A5

The alternating group of degree five is a p-normal group for all primes p, because all its Sylow subgroups are abelian. On the other hand, since the group is simple non-abelian, it is not p-solvable for p = 2,3,5.