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In a recent paper of the first author and I. M. Isaacs it was shown that if m = m(G) is the maximal order of an abelian subgroup of the finite group G, then |G| divides m! ([AI18, Thm. 5.2]). The purpose of this brief note is to improve on the m! bound (see Theorem 2.1 below). We shall then take up the task of determining when the (implicit) inequality of our theorem becomes an equality. Despite, perhaps, first appearances this determination is not trivial. To accomplish it we shall establish a result (Theorem 2.3) of independent interest and we shall then see that Theorems 2.1 and 2.3 combine to further strengthen Theorem 2.1 (see Theorem 3.4).
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When studying subgroups of $Out(F_n)$, one often replaces a given subgroup $H$ with one of its finite index subgroups $H_0$ so that virtual properties of $H$ become actual properties of $H_0$. In many cases, the finite index subgroup is $H_0 = H cap IA_n(Z/3)$. For which properties is this a good choice? Our main theorem states that being abelian is such a property. Namely, every virtually abelian subgroup of $IA_n(Z/3)$ is abelian.
In 1933 B.~H.~Neumann constructed uncountably many subgroups of ${rm SL}_2(mathbb Z)$ which act regularly on the primitive elements of $mathbb Z^2$. As pointed out by Magnus, their images in the modular group ${rm PSL}_2(mathbb Z)cong C_3*C_2$ are maximal nonparabolic subgroups, that is, maximal with respect to containing no parabolic elements. We strengthen and extend this result by giving a simple construction using planar maps to show that for all integers $pge 3$, $qge 2$ the triangle group $Gamma=Delta(p,q,infty)cong C_p*C_q$ has uncountably many conjugacy classes of nonparabolic maximal subgroups. We also extend results of Tretkoff and of Brenner and Lyndon for the modular group by constructing uncountably many conjugacy classes of such subgroups of $Gamma$ which do not arise from Neumanns original method. These maximal subgroups are all generated by elliptic elements, of finite order, but a similar construction yields uncountably many conjugacy classes of torsion-free maximal subgroups of the Hecke groups $C_p*C_2$ for odd $pge 3$. Finally, an adaptation of work of Conder yields uncountably many conjugacy classes of maximal subgroups of $Delta(2,3,r)$ for all $rge 7$.
Let $mathfrak{X}$ be a class of finite groups closed under taking subgroups, homomorphic images and extensions. It is known that if $A$ is a normal subgroup of a finite group $G$ then the image of an $mathfrak{X}$-maximal subgroup $H$ of $G$ in $G/A$ is not, in general, $mathfrak{X}$-maximal in $G/A$. We say that the reduction $mathfrak{X}$-theorem holds for a finite group $A$ if, for every finite group $G$ that is an extension of $A$ (i. e. contains $A$ as a normal subgroup), the number of conjugacy classes of $mathfrak{X}$-maximal subgroups in $G$ and $G/A$ is the same. The reduction $mathfrak{X}$-theorem for $A$ implies that $HA/A$ is $mathfrak{X}$-maximal in $G/A$ for every extension $G$ of $A$ and every $mathfrak{X}$-maximal subgroup $H$ of $G$. In this paper, we prove that the reduction $mathfrak{X}$-theorem holds for $A$ if and only if all $mathfrak{X}$-maximal subgroups are conjugate in $A$ and classify the finite groups with this property in terms of composition factors.
In this paper, we study a group in which every 2-maximal subgroup is a Hall subgroup.
Motivated in part by representation theoretic questions, we prove that if G is a finite quasi-simple group, then there exists an elementary abelian subgroup of G that intersects every conjugacy class of involutions of G.
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