In this article, we study rational matrix representations of VZ p-groups (p is any prime). Using our findings on VZ p-groups, we explicitly obtain all inequivalent irreducible rational matrix representations of all p-groups of order $\leq p^4$. Furthermore, we establish combinatorial formulae to determine the Wedderburn decompositions of rational group algebras for VZ p-groups and all p-groups of order $\leq p^4$, ensuring simplicity in the process.

Let p be an odd prime and let G be a non-abelian finite p-group of exponent p2 with three distinct characteristic subgroups, namely 1, Gp and G. The quotient group G/Gp gives rise to an anti-commutative 𝔽p-algebra L such that the action of Aut (L) is irreducible on L; we call such an algebra IAC. This paper establishes a duality G ↔ L between such groups and such IAC algebras. We prove that IAC algebras are semisimple and we classify the simple IAC algebras of dimension at most 4 over certain fields. We also give other examples of simple IAC algebras, including a family related to the m-th symmetric power of the natural module of SL(2, 𝔽).

We introduce the concept of infinite cochain sequences and initiate a theory of homological algebra for them. We show how these sequences simplify and improve the construction of infinite coclass families (as introduced by Eick and Leedham-Green) and also how they can be applied to prove that almost all groups in such a family have equivalent Quillen categories. We also include some examples of infinite families of $p$-groups from different coclass families that have equivalent Quillen categories.

In the 2006 edition of the Kourovka Notebook, Berkovich poses the following problem (Problem 16.13): Let $p$ be a prime and $P$ be a finite $p$-group. Can $P$ have every maximal subgroup special? We show that the structure of such groups is very restricted, but for all primes there are groups of arbitrarily large size with this property.

We obtain a sharp bound for p-elementary subgroups in the Cremona group Cr2(k) over an arbitrary perfect field k.

Let G be a nonabelian finite p-group of order pm. A long-standing conjecture asserts that G admits a noninner automorphism of order p. In this paper we prove the validity of the conjecture if exp (G)=pm−2. We also show that if G is a finite p-group of maximal class, then G has at least p(p−1) noninner automorphisms of order p which fix Φ(G) elementwise.

An old question of Brauer that asks how fast numbers of conjugacy classes grow is investigated by considering the least number cn of conjugacy classes in a group of order 2n. The numbers cn are computed for n ≤ 14 and a lower bound is given for c15. It is observed that cn grows very slowly except for occasional large jumps corresponding to an increase in coclass of the minimal groups Gn. Restricting to groups that are 2-generated or have coclass at most 3 allows us to extend these computations.

We make several conjectures, and prove some results, pertaining to conjugacy classes of a given size in finite groups, especially in p-groups and 2-groups.

We present a new algorithm which uses a cohomological approach to determine the groups of order pn, where p is a prime. We develop two methods to enumerate p-groups using the Cauchy-Frobenius Lemma. As an application we show that there are 10 494213 groups of order 29.

1991 Mathematics subject classification (Amer. Math. Soc.): primary 20D15.