Glauberman's replacement theorem

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This article defines a replacement theorem
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This article states and (possibly) proves a fact that is true for odd-order p-groups: groups of prime power order where the underlying prime is odd. The statement is false, in general, for groups whose order is a power of two.
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Statement

Suppose p is an odd prime, and P is a p-group. Let \mathcal{A}(P) be the set of abelian subgroups of maximum order in P and J(P) be the join of abelian subgroups of maximum order: the subgroup of P generated by the members of \mathcal{A}(P).

Suppose B is a class two normal subgroup of P such that its derived subgroup is contained in the center of J(P) (this center is also called the ZJ-subgroup of P) in symbols:

[B,B] \le Z(J(P)).

If A \in \mathcal{A}(P) is such that B does not normalize A, there exists A^* \in \mathcal{A}(P) such that:

  • A \cap B is a proper subgroup of A^* \cap B.
  • A^* normalizes A.

Related facts

Breakdown at the prime two

Other replacement theorems

For a complete list of replacement theorems, refer:

Category:Replacement theorems

Applications

References

Textbook references

Journal references