# Thompson's replacement theorem for elementary abelian subgroups

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

### For an odd prime

Suppose is a group of prime power order for an odd prime .

Let denote the set of all elementary abelian subgroups of maximum order in (i.e., for all elementary abelian subgroups of ).

Suppose and is an abelian subgroup such that normalizes but does not normalize . Then, there exists an elementary abelian subgroup of such that:

- , so in particular, .
- is a proper subgroup of .
- normalizes .

### For the prime

A slight modification works for the prime , but we have to drop the requirement that be elementary abelian and instead only have abelian but of size at least that of :

Let denote the set of all elementary abelian subgroups of maximum order in (i.e., for all elementary abelian subgroups of ).

Suppose and is an abelian subgroup such that normalizes but does not normalize . Then, there exists an abelian subgroup of such that:

- .
- is a proper subgroup of .
- normalizes .