In this paper we determine all finite $2$-groups of class $2$ in which every automorphism of order $2$ leaving the Frattini subgroup elementwise fixed is inner.

In this paper we determine all finite $2$-groups of class $2$ in which every automorphism of order $2$ leaving the Frattini subgroup elementwise fixed is inner.

For any given finite abelian group, we give factorizations of the group determinant in the group algebra of any subgroups. The factorizations is an extension of Dedekind's theorem. The extension leads to a generalization of Dedekind's theorem.

A finite group $G$ satisfies the on-prime power hypothesis for conjugacy class sizes if any two conjugacy class sizes $m$ and $n$ are either equal or have a common divisor a prime power. Taeri conjectured that an insoluble group satisfying this condition is isomorphic to $S times A$ where $A$ is abelian and $S cong PSL_2(q)$ for $q in {4,8}$. We confirm this conjecture.

We introduce right amenability, right FØlner nets, and right paradoxical decompositions for left homogeneous spaces and prove the Tarski-FØlner theorem for left homogeneous spaces with finite stabilisers. It states that right amenability, the existence of right FØlner nets, and the non-existence of right paradoxical decompositions are equivalent.

We determine a new infinite sequence of finite $2$-groups with deficiency zero. The groups have $2$ generators and $2$ relations, they have coclass $3$ and they are not metacyclic.