finite group theory In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...

, the Higman–Sims asymptotic formula gives an asymptotic estimate on number of groups of prime power order.

Statement

Let

p

be a (fixed)

prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

. Define

f(n,p)

as the number of isomorphism classes of groups of order

p^n

. Then: :

f(n,p) = p^

Here, the

big-O notation Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by German mathematicians Pau ...

is with respect to

n

, not with respect to

p

(the constant under the big-O notation may depend on

p

References

* * * Group theory Theorems in group theory {{group-theory-stub