For each prime $p$ we construct a family $\{G_{i}\}$ of finite $p$-groups such that $|\text{Aut}(G_{i})|/|G_{i}|$ tends to zero as $i$ tends to infinity. This disproves a well-known conjecture that $|G|$ divides $|\text{Aut}(G)|$ for every nonabelian finite $p$-group $G$.

Let ℓ be a prime number. It is not known whether every finite ℓ-group of rank n≥1 can be realized as a Galois group over with no more than n ramified primes. We prove that this can be done for the (minimal) family of finite ℓ-groups which contains all the cyclic groups of ℓ-power order and is closed under direct products, (regular) wreath products and rank-preserving homomorphic images. This family contains the Sylow ℓ-subgroups of the symmetric groups and of the classical groups over finite fields of characteristic not ℓ. On the other hand, it does not contain all finite ℓ-groups.

In this study of the behaviour of the number of conjugacy classes in finite $p$-groups using pro-$p$ groups, the conjugacy growth function $r_n(G)= \max\{r(G/N)\,{|}\,N\triangleleft_o G,|G\,{:}\,N|=n\}$ is introduced. It is proved that there are no infinite pro-$p$ groups of linear conjugacy growth (that is, there is no $c$ such that $r_n(G)\le c\log_2 n$ for all $n>1$) and it is shown that many known pro-$p$ groups are of exponential conjugacy growth (that is, there exists $\epsilon\,{>}\,0$ such that $r_n(G)\,{\ge}\,n^\epsilon$ for infinitely many values of $n$).