Let $p$ be an odd prime. We construct a $p$-group $P$ of nilpotency class two, rank seven and exponent $p$, such that $\text{Aut}(P)$ induces $N_{\text{GL}(7,p)}(G_{2}(p))=Z(\text{GL}(7,p))G_{2}(p)$ on the Frattini quotient $P/\unicode[STIX]{x1D6F7}(P)$. The constructed group $P$ is the smallest $p$-group with these properties, having order $p^{14}$, and when $p=3$ our construction gives two nonisomorphic $p$-groups. To show that $P$ satisfies the specified properties, we study the action of $G_{2}(q)$ on the octonion algebra over $\mathbb{F}_{q}$, for each power $q$ of $p$, and explore the reducibility of the exterior square of each irreducible seven-dimensional $\mathbb{F}_{q}[G_{2}(q)]$-module.

Let $G$ be a $p$-group and let $\unicode[STIX]{x1D712}$ be an irreducible character of $G$. The codegree of $\unicode[STIX]{x1D712}$ is given by $|G:\,\text{ker}(\unicode[STIX]{x1D712})|/\unicode[STIX]{x1D712}(1)$. If $G$ is a maximal class $p$-group that is normally monomial or has at most three character degrees, then the codegrees of $G$ are consecutive powers of $p$. If $|G|=p^{n}$ and $G$ has consecutive $p$-power codegrees up to $p^{n-1}$, then the nilpotence class of $G$ is at most 2 or $G$ has maximal class.

If the centre of a group $G$ is trivial, then so is the centre of its automorphism group. We study the structure of the centre of the automorphism group of a group $G$ when the centre of $G$ is a cyclic group. In particular, it is shown that the exponent of $Z(\text{Aut}(G))$ is less than or equal to the exponent of $Z(G)$ in this case.

Bob Oliver conjectures that if p is an odd prime and S is a finite p-group, then the Oliver subgroup contains the Thompson subgroup Je(S). A positive resolution of this conjecture would give the existence and uniqueness of centric linking systems for fusion systems at odd primes. Using the ideas and work of Glauberman, we prove that if p ≥ 5, G is a finite p-group, and V is an elementary abelian p-group which is an F-module for G, then there exists a quadratic offender which is 2-subnormal (normal in its normal closure) in G. We apply this to show that Oliver's Conjecture holds provided that the quotient has class at most log2(p − 2) + 1, or p ≥ 5 and G is equal to its own Baumann subgroup.

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.

We show that if G is any p-group of class at most two and exponent p, then there exist groups G1 and G2 of class two and exponent p that contain G, neither of which can be expressed as a central product, and with G1 capable and G2 not capable. We provide upper bounds for rank(Giab) in terms of rank(Gab) in each case.

We show that an elation generalized quadrangle that has p+1 lines on each point, for some prime p, is classical or arises from a flock of a quadratic cone (that is, is a flock quadrangle).

We show that p-groups of order p5 are determined by their group algebras over the field of p elements. Many cases have been dealt with in earlier work of ourselves and others. The only case whose details remain to be given here is that of groups of nilpotency class 3 for p odd.

We show that p-groups of maximal class and order p5 are determined by their group algebras over the field of p elements. The most important information requisite for the proof is obtained from a detailed study of the unit group of a quotient algebra of the group algebra, larger than the small group algebra.

The unit group of the modular group algebra of a finite p-group in characteristic p is nilpotent. The p-groups for which it is of nilpotency class 3 were determined in work of Coleman, Passman, Shalev and Mann when p ≥ 3. We resolve the p = 2 case here which completes the classification.