Which finite simple groups are unit groups?
Christopher Davis
Tommy Occhipinti
We prove that if $G$ is a finite simple group which is the unit group of a ring, then $G$ is isomorphic to either (a) a cyclic group of order 2; (b) a cyclic group of prime order $2^k -1$ for some $k$; or (c) a projective special linear group $\PSL_n(\FF_2)$ for some $n \geq 3$. Moreover, these groups do (trivially) all occur as unit groups. We deduce this classification from a more general result, which holds for groups $G$ with no non-trivial normal 2-subgroup.
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\author{Christopher Davis}
\address{University of California, Irvine, Dept of
Mathematics, Irvine, CA 92697}
\email{davis@math.uci.edu}
\date{\today}

\author{Tommy Occhipinti}
\address{University of California, Irvine, Dept of
Mathematics, Irvine, CA 92697}
\email{tocchipi@math.uci.edu}



\title{Which finite simple groups are unit groups?}

\begin{document}

\begin{abstract}
We prove that
if $G$ is a finite simple group which is the unit group of a ring, then $G$ is isomorphic to either (a) a cyclic group of order 2; (b) a cyclic group of prime order $2^k -1$ for some $k$; or (c) a projective special linear group $\PSL_n(\FF_2)$ for some $n \geq 3$.  Moreover, these groups do (trivially) all occur as unit groups.   We deduce this classification from a more general result, which holds for groups $G$ with no non-trivial normal 2-subgroup.
\end{abstract}


\maketitle


The finite groups $G$ of odd order which occur as unit groups of rings\footnote{Our rings are assumed unital but not necessarily commutative, and ring homomorphisms send $1$ to $1$.}  were determined in \cite{Dit71}.  We will prove similar results for a more general class of groups; the description of this class of groups uses the following.    

\begin{Def}
For a finite group $G$, the \emph{$p$-core} of $G$ is the largest normal $p$-subgroup of $G$.  We denote this subgroup by $O_p(G)$.   It is the intersection of all Sylow $p$-subgroups of $G$.  
\end{Def}

We now state the main result.  The authors\footnote{The authors also thank Colin Adams, John F.\ Dillon, Dennis Eichhorn, Noam Elkies, Kiran Kedlaya, Charles Toll, and Ryan Vinroot for many useful discussions.} are most grateful to the anonymous referee for our earlier paper \cite{DO13}, who recognized that one of the results proved in that paper could be strengthened into the following. 

\begin{Thm} \label{main theorem}
Let $G$ denote a finite group such that $O_2(G) = \{1\}$ and such that $G$ is isomorphic to the unit group of a ring $R$.  Then 
\[
G \cong \GL_{n_1} (\FF_{2^{k_1}}) \times \cdots \times \GL_{n_r}(\FF_{2^{k_r}}).
\]
\end{Thm}




Before proving Theorem~\ref{main theorem}, we record the following corollary.


\begin{Cor}
The finite simple groups which occur as unit groups of rings are precisely the groups
\begin{enumerate}[label=(\alph*)]
\item $\ZZ/2\ZZ$,
\item $\ZZ/p\ZZ$ for a Mersenne prime $p = 2^{k}-1$,
\item $\PSL_n(\FF_2)$ for $n \geq 3$.
\end{enumerate}
\end{Cor}

\begin{proof}
If $G$ is a finite simple group, then either $O_2(G) = \{1\}$ or $O_2(G) = G$.  If $O_2(G) = G$, then $G$ is a $2$-group, and because we are assuming $G$ is simple, we must have $G \cong \ZZ/2\ZZ$, which for instance is isomorphic to the unit group of $\ZZ$.  

Hence assume $G$ is a finite simple group which is isomorphic to the unit group of a ring and further assume $O_2(G) = \{1\}$.   
By Theorem~\ref{main theorem}, we know
\[
G \cong \GL_{n_1}(\FF_{2^{k_1}}) \times \cdots \times \GL_{n_r}(\FF_{2^{k_r}}).
\]
Clearly these groups all occur as unit groups, so we are reduced to determining which of them are simple;  this forces
\[
G \cong \GL_n(\FF_{2^k}).
\]
If $n > 1$ and $k > 1$, then the subgroup of invertible scalar matrices forms a nontrivial normal subgroup.  Hence two possibilities remain.  If $n = 1$, then $\GL_1(\FF_{2^k})$ is cyclic of order $2^k-1$.  If $k =1$, then $\GL_n(\FF_2) = \PSL_n(\FF_2)$.  This completes the proof.  
\end{proof}

\begin{Rmk}
The simple groups $A_8$ and $\PSL_2(\FF_7)$ also occur as unit groups.  This follows immediately from the exceptional isomorphisms
\[
A_8 \cong \PSL_4(\FF_2) \text{ and } \PSL_2(\FF_7) \cong \PSL_3(\FF_2).
\]
See for instance \cite[Section~3.12]{Wil09}.
\end{Rmk}

Having recorded the above consequences of the main result, we now gather the preliminary results used in its proof.  We begin with the following easy exercise.  

\begin{Lem} \label{trivial 2-core implies characteristic 2}
Let $G$ denote a finite group with $O_2(G) = \{1\}$, and let $R$ denote a ring with $R^{\times} \cong G$.  Then $R$ has characteristic 2.
\end{Lem}

\begin{proof}
The elements $1$ and $-1$ are units in $R$ and are in the center of $R$, hence are in the center of $R^{\times}$.  By the assumption $O_2(G) = \{1\}$, the center of $G$ cannot contain any elements of order~2.  Hence $1 = -1$.
\end{proof}

\begin{Lem} \label{ideal exists}
Keep notation as in Lemma~\ref{trivial 2-core implies characteristic 2}, and fix an isomorphism $R^{\times} \cong G$.  
The image of the natural map
\[\varphi: \FF_2[G] \rightarrow R\]
is a ring with unit group isomorphic to $G$.  
\end{Lem}

\begin{proof}
Write $S$ for the image of $\varphi$.  It is clear on one hand that $S^{\times} \subseteq R^{\times} \cong G$, and on the other hand it is clear that the induced map $\varphi: G \rightarrow S^{\times} \rightarrow R^{\times}$ is surjective.  
\end{proof}

\begin{Lem} \label{square zero ideal}
Let $R$ denote a finite ring of characteristic~2.  If $J \subseteq R$ is a two-sided ideal such that $J^2 = 0$, then $1 + J$ is a normal elementary abelian $2$-subgroup of $R^{\times}$.
\end{Lem}

\begin{proof}
Note that for any $j \in J$ and $r \in R^{\times}$, we have $(1 + j)^2 = 1 + j^2 = 1$ and 
\[r(1+j)r^{-1} = 1 + rjr^{-1} \in 1 + J.\]
\end{proof}

We now use these preliminary results to prove our main theorem. 

\begin{proof}[Proof of Theorem~\ref{main theorem}]
By Lemma~\ref{ideal exists}, we may assume $R$ is a finite ring (and is in particular artinian) and has characteristic~2.  Let $J$ denote a two-sided ideal of $R$ such that $J^2 = 0$.  By Lemma~\ref{square zero ideal}, the set $1 + J$ is a normal $2$-subgroup of $R^{\times}$, and so by the assumption $O_2(G) = \{1\}$, we have $J = \{0\}$.   Thus the ring $R$ has no non-zero two-sided ideals $J$ with $J^2 = 0$, and hence $R$ has no non-zero two-sided nilpotent ideals.  By \cite[Theorem~5.4.5]{Coh89}, the artinian ring $R$ is semisimple.    By Wedderburn's Theorem \cite[Theorem~5.3.4]{Coh89}, we have
\[
R \cong M_{n_1}(D_1) \times \cdots \times M_{n_r}(D_r)
\]
for some $n_1, \ldots, n_r \geq 1$ and some division algebras $D_1, \ldots, D_r$.  Our ring $R$ is finite and hence each $D_i$ is finite. By another theorem of Wedderburn \cite[Theorem~3.8.6]{Coh89}, we have that each $D_i$ is a finite field.  Finally, because the ring $R$ has characteristic~2, each field $D_i$ has characteristic~2.  This completes the proof.
\end{proof}



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\end{document}












We classify the finite groups $G$ with $O_2(G) = \{1\}$ which can occur as unit groups of rings.  Before stating our main result, we outline the proof.  Let $G$ denote a finite group with $O_2(G) = \{1\}$, and let $R$ denote a ring with unit group isomorphic to $G$.  We first prove that $R$ must have characteristic~2, and then that the image of $\FF_2[G]$ in $R$ must also have unit group isomorphic to $G$.  By showing that the image of $\FF_2[G]$ in $R$ is semisimple, we are able to determine the possibilities for $G$ using Wedderburn's Theorem.


\subsection*{Acknowledgments}
This paper grew out of an earlier project that determined which alternating and symmetric groups were unit groups of rings \cite{DO13}.  The authors are most grateful to the anonymous referee for that paper, who recognized that our earlier result could be strengthened into Theorem~\ref{main theorem} above.  We would also like to thank Colin Adams, John F.\ Dillon, Dennis Eichhorn, Noam Elkies, Kiran Kedlaya, and Ryan Vinroot for many useful conversations.  Special thanks also to Charles Toll, who first proposed this project to the second author.


simple.bbl
\begin{thebibliography}{1}

\bibitem{Coh89}
P.~M. Cohn.
\newblock {\em Algebra. {V}ol. 2}.
\newblock John Wiley \& Sons Ltd., Chichester, second edition, 1989.

\bibitem{DO13}
Christopher Davis and Tommy Occhipinti.
\newblock Which alternating and symmetric groups are unit groups?
\newblock {\tt http://arxiv.org/abs/1304.0143}.

\bibitem{Dit71}
S.~Z. Ditor.
\newblock On the group of units of a ring.
\newblock {\em Amer. Math. Monthly}, 78:522--523, 1971.

\bibitem{Wil09}
R.~Wilson.
\newblock {\em The Finite Simple Groups}.
\newblock Graduate Texts in Mathematics. Springer, 2009.

\end{thebibliography}
