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Robert A. Wilson
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There are four finite groups that could plausibly play the role of the spin group in a finite or discrete model of quantum mechanics, namely the four double covers of the three rotation groups of the Platonic solids. In an earlier paper I have considered in detail how the smallest of these groups, namely the binary tetrahedral group, of order 24, could give rise to a non-relativistic theory that contains much of the structure of the standard model of particle physics. In this paper I consider how one of the two double covers of the rotation group of the cube might extend this to a relativistic theory.
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A group $G$ is said to be $frac{3}{2}$-generated if every nontrivial element belongs to a generating pair. It is easy to see that if $G$ has this property then every proper quotient of $G$ is cyclic. In this paper we prove that the converse is true for finite groups, which settles a conjecture of Breuer, Guralnick and Kantor from 2008. In fact, we prove a much stronger result, which solves a problem posed by Brenner and Wiegold in 1975. Namely, if $G$ is a finite group and every proper quotient of $G$ is cyclic, then for any pair of nontrivial elements $x_1,x_2 in G$, there exists $y in G$ such that $G = langle x_1, y rangle = langle x_2, y rangle$. In other words, $s(G) geqslant 2$, where $s(G)$ is the spread of $G$. Moreover, if $u(G)$ denotes the more restrictive uniform spread of $G$, then we can completely characterise the finite groups $G$ with $u(G) = 0$ and $u(G)=1$. To prove these results, we first establish a reduction to almost simple groups. For simple groups, the result was proved by Guralnick and Kantor in 2000 using probabilistic methods and since then the almost simple groups have been the subject of several papers. By combining our reduction theorem and this earlier work, it remains to handle the groups whose socles are exceptional groups of Lie type and this is the case we treat in this paper.
Let $G$ be a finite group admitting a coprime automorphism $phi$ of order $n$. Denote by $G_{phi}$ the centralizer of $phi$ in $G$ and by $G_{-phi}$ the set ${ x^{-1}x^{phi}; xin G}$. We prove the following results. 1. If every element from $G_{phi}cup G_{-phi}$ is contained in a $phi$-invariant subgroup of exponent dividing $e$, then the exponent of $G$ is $(e,n)$-bounded. 2. Suppose that $G_{phi}$ is nilpotent of class $c$. If $x^{e}=1$ for each $x in G_{-phi}$ and any two elements of $G_{-phi}$ are contained in a $phi$-invariant soluble subgroup of derived length $d$, then the exponent of $[G,phi]$ is bounded in terms of $c,d,e,n$.
We study the class of finite groups $G$ satisfying $Phi (G/N)= Phi(G)N/N$ for all normal subgroups $N$ of $G$. As a consequence of our main results we extend and amplify a theorem of Doerk concerning this class from the soluble universe to all finite groups and answer in the affirmative a long-standing question of Christensen whether the class of finite groups which possess complements for each of their normal subgroups is subnormally closed.
Assume that $G$ is a finite group. For every $a, b inmathbb N,$ we define a graph $Gamma_{a,b}(G)$ whose vertices correspond to the elements of $G^acup G^b$ and in which two tuples $(x_1,dots,x_a)$ and $(y_1,dots,y_b)$ are adjacent if and only if $langle x_1,dots,x_a,y_1,dots,y_b rangle =G.$ We study several properties of these graphs (isolated vertices, loops, connectivity, diameter of the connected components) and we investigate the relations between their properties and the group structure, with the aim of understanding which information about $G$ are encoded by these graphs.
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