# Congruence condition on number of abelian subgroups of small prime power order and bounded exponent for odd prime

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This article is about a congruence condition.

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## Contents

## Statement

### In terms of universal congruence condition

Suppose is an odd prime, and . Then, the collection of abelian groups of order and exponent dividing is a Collection of groups satisfying a universal congruence condition (?).

### Hands-on statement

Suppose is an odd prime, and is a finite -group. Suppose . Suppose has an abelian subgroup of order and exponent dividing . Then, the following equivalent statements hold:

- The number of abelian subgroups of of order and exponent dividing is either equal to zero or congruent to modulo .
- The number of abelian normal subgroups of of order and exponent dividing is congruent to modulo .
- For any finite -group containing , the number of abelian subgroups of of order and exponent dividing that are normal in is congruent to modulo .