<p>In this article, we determine the non-real elements—the ones that are not conjugate to their inverses—in the group <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2024_801_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="76" /> </InlineMediaObject> <EquationSource Format="TEX">$G=\textrm{G}_2(q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>=</mo> <msub> <mtext>G</mtext> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> when <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2024_801_Article_IEq4.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="106" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{char}\hspace{0.55542pt}({\mathbb {F}}_q)\ne 2, 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>char</mtext> <mspace width="0.55542pt" /> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mo stretchy="false">)</mo> <mo>≠</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. We use this to find when this group is chiral; that is, there is a word <i>w</i> such that <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2024_801_Article_IEq5.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="115" /> </InlineMediaObject> <EquationSource Format="TEX">$w(G)\ne w(G)^{-1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>w</mi> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo>≠</mo> <mi>w</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>. We also show that most classical finite simple groups are achiral.</p>

Chirality and non-real elements in $\textrm{G}_2(q)$

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发表日期：

2025年1月8日

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Sushil Bhunia,
Amit Kulshrestha,
Anupam Singh

摘要

In this article, we determine the non-real elements—the ones that are not conjugate to their inverses—in the group $G=\textrm{G}_2(q)$ $G = G_{2} (q)$ when $\textrm{char}\hspace{0.55542pt}({\mathbb {F}}_q)\ne 2, 3$ $char (F_{q}) \neq 2, 3$ . We use this to find when this group is chiral; that is, there is a word w such that $w(G)\ne w(G)^{-1}$ $w (G) \neq w {(G)}^{- 1}$ . We also show that most classical finite simple groups are achiral.

Chirality and non-real elements in \(\textrm{G}_2(q)\)

摘要