# Minimum size of generating set is bounded by sum of exponents of prime divisors of order

From Groupprops

## Statement

Suppose is a finite group of order , which has prime factorization:

Then, the minimum size of generating set for is at most equal to:

Moreover, equality holds if and only if is a direct product of elementary abelian groups of order , i.e.:

## Facts used

- Any generating set of minimum size must be an minimal generating set (in the sense of being irredundant).
- Size of minimal generating set is bounded by max-length
- Max-length is bounded by sum of exponents of prime divisors of order

## Proof

The proof follows directly by combining Facts (1)-(3).