Revisit the first problem set

Definition 1.1 A product-free set in a group

is a subset $L\subset G$ such that the equation

has no solution in

.
For a finite group

, let $\alpha(G)$ be the proportion of

in the largest product free set; $\alpha(G)=\vert L\vert/\vert G\vert$ , for $L\subset G$ the largest product-free set.

$\begin{exercise} % latex2html id marker 812If $G\to H$\ is a surjective homomorphism, show that $\alpha(G)\ge\alpha(H)$. \end{exercise}$

$\begin{exercise} % latex2html id marker 814 $\alpha(G)\le \frac12$. \end{exercise}$

$\begin{exercise} % latex2html id marker 816 If $G$\ has a subgroup of index $2$,... ...$G$\ is an abelian groups of even order then $\alpha(G)=\frac12$. \end{exercise}$

$\begin{exercise} % latex2html id marker 818 If $G=K\times L$\ then $\alpha(G)=\max\{\alpha(K),\alpha(L)\}$. \end{exercise}$

Definition 1.2 A group is finitely generated if it has a finite set of generators.

$\begin{exercise} % latex2html id marker 820Find an infinite group $G$\ such th... ...e multiplicative group of complex numbers of unit absolute value. \end{exercise}$

$\begin{exercise} % latex2html id marker 822Let $G$\ be an infinite abelian gro... ...is a subgroup. ($H$\ is called the {\bf torsion subgroup} of $G$. \end{exercise}$

Theorem 1.3 (Fundamental theorem of finitely generated abelian groups) Every finitely generated abelian group is the direct product of a finite number of cyclic groups. The number of infinite cycic groups in this factorization is unique. The product of the finite abelian subgroups in this factorization is unique; it is the torsion subgroup of

$\begin{exercise} % latex2html id marker 824Let $G$\ be a finitely generated ab... ...oup $L$\ is not unique, unless $G$\ is \lq\lq torsion-free'' ($T=\{1\}$\end{exercise}$

$\begin{exercise} % latex2html id marker 826 Prove that $K\times L$\ is, the dire... ...roup is the direct product of cyclic groups of prime power order. \end{exercise}$

Theorem 1.4 (Fundamental theorem of finite abelian groups) Every finite abelian group is the direct product of cyclic groups of prime power order. The orders in this factorization are unique.

$\begin{exercise} Show that the subgroups in this factorization need not be unique. \end{exercise}$

Thus to compute $\alpha(G)$ for finite abelian groups, it suffices to know $\alpha({\mathbb{Z}}_{p^k})$ .

Corollary 1.5 (Fundamental theorem of finite abelian groups, Smith normal form) Every finite abelian group is the direct product of cyclic groups ${\mathbb{Z}}_{n_1}\times\dots{\mathbb{Z}}_{n_k}$ where $2\le n_1$ and $n_{i-1}\mid n_i$ for all

. The values $n_1,\dots,n_k$ are unique.

$\begin{exercise} % latex2html id marker 833 $\alpha({\mathbb{Z}}_7)=2/7$, contrary to the lower bound $\frac13$\ claimed in the first problem set. \end{exercise}$

$\begin{exx} % latex2html id marker 836 {\bf (Cauchy-Davenport)} For a prime $p$,... ...\in B\}$. Then $\vert A+B\vert=\max\{p,\vert A\vert+\vert B\vert-1\}$. \end{exx}$

$\begin{exercise} % latex2html id marker 840 $\alpha({\mathbb{Z}}_p)=\lfloor\frac... ... the group. Use the Cauchy-Davenport Theorem for the upper bound. \end{exercise}$

$\begin{exercise} % latex2html id marker 843 If $G$\ is abelian then $\frac27\le \alpha(G)\le\frac12$. \end{exercise}$

$\begin{exercise} % latex2html id marker 845 What is the exact value of $\alpha({... ...\lq sum-free sets'' (abelian groups are comonly written additively). \end{exercise}$

But the real question is for nonabelian groups. The most interesting cases would be classes of simple groups, starting with the alternating groups and the projective special linear groups.

$\begin{exercise} % latex2html id marker 848 If $H\lneq G$\ then $\alpha(G)\ge\frac{1}{[G:H]}$. {\em Hint.} \ take a coset. \end{exercise}$

$\begin{exercise} % latex2html id marker 850 $\alpha(A_n)\ge\frac1n$. \end{exercise}$

Conjecture 1.6 $\alpha(A_n)=o(1)$ (i.e., $\lim_{n\to\infty}\alpha(A_n)=0$ .)

Nothing better than the inequalities $1/n\le \alpha(A_n)\le 1/2$ appears to be known for $n\ge 5$ . Perhaps $\alpha(A_n)=1/n$ but the instructor would not bet on this one.

Conjecture 1.7 Let

be an infinite sequence of finite simple groups ( $\vert G_n\vert\to\infty$ ). Then $\alpha(G_n)=o(1)$ .

It would be of interest to prove this various inifinite classes of finite simple groups, such as the projective special linear groups.