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There are several results in the literature concerning $p$-groups $G$ with a maximal elementary abelian normal subgroup of rank $k$ due to Thompson, Mann and others. Following an idea of Sambale we obtain bounds for the number of generators etc. of a $2$-group $G$ in terms of $k$, which were previously known only for $p>2$. We also prove a theorem that is new even for odd primes. Namely, we show that if $G$ has a maximal elementary abelian normal subgroup of rank $k$, then for any abelian subgroup $A$ the Frattini subgroup $\Phi(A)$ can be generated by $2k$ elements ($3k$ when $p=2$). The proof of this rests upon the following result of independent interest: If $V$ is an $n$-dimensional vector space, then any commutative subalgebra of End$(V)$ contains a zero algebra of codimension at most $n$., Comment: 11 pages. Some minor errors has been corrected

The minimal base size $b(G)$ for a permutation group $G$, is a widely studied topic in the permutation group theory. Z. Halasi and K. Podoski proved that $b(G)\leq 2$ for coprime linear groups. Motivated by this result and the probabilistic method used by T. C. Burness, M. W. Liebeck and A. Shalev, it was asked by L. Pyber that for coprime linear groups $G\leq GL(V)$, whether there exists a constant $c$ such that the probability of that a random $c$-tuple is a base for $G$ tends to 1 as $|V|\to\infty$. While the answer to this question is negative in general, it is positive under the additional assumption that $G$ is even primitive as a linear group. In this paper, we show that almost all $11$-tuples are bases for coprime primitive linear groups.

The flow semigroup, introduced by John Rhodes, is an invariant for digraphs and a complete invariant for graphs. After collecting together previous partial results, we refine and prove Rhodes's conjecture on the structure of the maximal groups in the flow semigroup for finite, antisymmetric, strongly connected digraphs. Building on this result, we investigate and fully describe the structure and actions of the maximal subgroups of the flow semigroup acting on all but $k$ points for all finite digraphs and graphs for all $k\geq 1$. A linear algorithm (in the number of edges) is presented to determine these so-called `defect $k$ groups' for any finite (di)graph. Finally, we prove that the complexity of the flow semigroup of a 2-vertex connected (and strongly connected di)graph with $n$ vertices is $n-2$, completely confirming Rhodes's conjecture for such (di)graphs., Comment: 25 pages

Let G be a linear group acting on the finite vector space V and assume that (|G|,|V|)=1. In this paper we prove that G has a base size at most two and this estimate is sharp. This generalizes and strengthens several former results concerning base sizes of coprime linear groups. As a direct consequence, we answer a question of I. M. Isaacs in the affirmative. Via large orbits this is related to the k(GV) theorem., Comment: Minor corrections. Theorem 5.9 and Remark 5.15 have been rephrased in a more general form

Let $G$ be a solvable linear group acting on the finite vectorpace $V$ and assume that $(|G|,|V|)=1$. In this paper we find $x,y\in V$ such that $C_G(x)\cap C_G(y)=1$. In particular, this answers a question of I. M. Isaacs. We complete some results of S. Dolphi, A. Seress and T. R. Wolf.

Let $G$ be a finite group with derived subgroup of rank $r$. We prove that $\gzz\leq |G'|^{2r}$. Motivated by the results of I. M. Isaacs in \cite{isa} we show that if $G$ is capable then $\gz\leq |G'|^{4r}$. This answers a question of L. Pyber. We prove that if $G$ is a capable $p$-group then the rank of $G/\mathbf{Z}(G)$ is bounded above in terms of the rank of $G'$.