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proof that a nontrivial normal subgroup of a finite -group and the center of have nontrivial intersection
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(Proof)
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Define to act on by conjugation; that is, for , , define
Note that
since
. This is easily seen to be a well-defined group action.
Now, the set of invariants of under this action are
The class equation theorem states that
where the are proper subgroups of , and thus that
We now use elementary group theory to show that divides each term on the right, and conclude as a result that divides
, so that
cannot be trivial.
As is a nontrivial finite -group, it is obvious from Cauchy's theorem that for . Since and the are subgroups of , each either is trivial or has order a power of , by Lagrange's theorem. Since is nontrivial, its order is a nonzero power of . Since each is a proper subgroup of and has order a power of , it follows that
also has order a nonzero power of .
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"proof that a nontrivial normal subgroup of a finite -group and the center of have nontrivial intersection" is owned by rm50. [ full author list (2) | owner history (1) ]
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Cross-references: Lagrange's theorem, power, order, subgroups, Cauchy's theorem, obvious, finite, right, term, divides, theory, group, proper subgroups, class equation theorem, action, invariants, group action, well-defined, conjugation, act on
There is 1 reference to this entry.
This is version 6 of proof that a nontrivial normal subgroup of a finite -group and the center of have nontrivial intersection, born on 2004-05-02, modified 2007-01-23.
Object id is 5828, canonical name is ProofOfANontrivialNormalSubgroupOfAFinitePGroupGAndTheCenterOfGHaveNontrivialIntersection.
Accessed 3501 times total.
Classification:
| AMS MSC: | 20D20 (Group theory and generalizations :: Abstract finite groups :: Sylow subgroups, Sylow properties, $\pi$-groups, $\pi$-structure) |
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Pending Errata and Addenda
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