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Images of word maps in finite simple groups. (English) Zbl 1318.20014

Let \(G\) be a group, and let \(w(a,b)\) be an element of \(F(a,b)\), the free group on two letters. Then we define the associated word map to be \[ \varphi_w\colon G\times G \to G,\qquad (g_1,g_2)\mapsto w(g_1,g_2). \] This paper answers a question of Shalev concerning the possible images of such word maps.
The main result is the following: Theorem: Let \(G\) be a finite simple group, and let \(A\) be a subset of \(G\). If \(1\in A\) and \(\alpha(A)=A\) for all \(\alpha\in\operatorname{Aut}(G)\), then there exists \(w(a,b)\in F(a,b)\) such that \(\text{Im}(\varphi_w)=A\).
Note that the converse of the theorem is straightforward. The proof given here depends on the Classification of Finite Simple Groups (CFSG) through its use of a result of Guralnick and Kantor. A natural question is whether a non-CFSG-dependent proof can be found.
A second result is presented which is of independent interest and which does not depend on CFSG:
Proposition: For every finite simple group \(G\), there is a word \(w(a,b)\in F(a,b)\) such that, for all \((x,y)\in G\times G\), \(w(x,y)\neq 1\) if and only \(\langle x,y\rangle=G\).
Both results are proved by considering the diagonal map \[ \varphi_M\colon F(a,b)\to G^{|G|^2},\qquad w\mapsto (w(a_1,b_1),w(a_2, b_2),\ldots) \] where \(\{(a_i,b_i)\mid i=1,\ldots,|G|^2\}\) is the set of all ordered pairs of elements of \(G\).

MSC:

20D05 Finite simple groups and their classification
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
20F70 Algebraic geometry over groups; equations over groups
20F05 Generators, relations, and presentations of groups
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References:

[1] DOI: 10.1017/CBO9781139107082 · doi:10.1017/CBO9781139107082
[2] DOI: 10.1515/jgt.2010.039 · Zbl 1234.20019 · doi:10.1515/jgt.2010.039
[3] DOI: 10.1006/jabr.2000.8357 · Zbl 0973.20012 · doi:10.1006/jabr.2000.8357
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