p-Group

A Finite Group of Order $p^a$ for $p$ a Prime is called a $p$-group. Sylow proved that every Group of this form has a Power-commutator representation on $n$ generators defined by

$$a_i^p =\prod_{k=i+1}^n a_k^{\beta(i,k)}$$ (1)

for $0\leq\beta(i,k)<p$, $1\leq i\leq n$ and

$$[a_j,a_i]=\prod_{k=j+1}^n a_k^{\beta(i,j,k)}$$ (2)

for $0\leq\beta(i,j,k)<p$, $1\leq i<j\leq n$. If $p$ is Prime and $f(p)$ the number of Groups of order $p^m$, then

$$f(p)=p^{Am^2},$$ (3)

where

$$\lim_{m\to\infty} A={\textstyle{2\over 27}}$$ (4)

(Higman 1960a,b).

See also Finite Group

References

Higman, G. ``Enumerating $p$-Groups. I. Inequalities.'' Proc. London Math. Soc. 10, 24-30, 1960a.

Higman, G. ``Enumerating $p$-Groups. II. Problems Whose Solution is PORC.'' Proc. London Math. Soc. 10, 566-582, 1960b.