Assume F is a finite field of order <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2114_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$p^f$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>p</mi> <mi>f</mi> </msup> </math></EquationSource> </InlineEquation> and q is an odd prime for which <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2114_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="97" /> </InlineMediaObject> <EquationSource Format="TEX">$p^f-1=sq^m$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>p</mi> <mi>f</mi> </msup> <mo>-</mo> <mn>1</mn> <mo>=</mo> <mi>s</mi> <msup> <mi>q</mi> <mi>m</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2114_Article_IEq3.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">$m \ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2114_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="67" /> </InlineMediaObject> <EquationSource Format="TEX">$(s,q)=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. In this article, we obtain the order of the symmetric and the unitary subgroup of the semisimple group algebra <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2114_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">$FC_q.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <msub> <mi>C</mi> <mi>q</mi> </msub> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> Further, for the extension G of <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2114_Article_IEq6.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">$C_q = \langle b \rangle $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mi>q</mi> </msub> <mo>=</mo> <mrow> <mo stretchy="false">⟨</mo> <mi>b</mi> <mo stretchy="false">⟩</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> by an abelian group A of order <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2114_Article_IEq7.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$p^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>p</mi> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2114_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="93" /> </InlineMediaObject> <EquationSource Format="TEX">$C_{A}(b) = \{e\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mi>A</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mrow> <mo stretchy="false">{</mo> <mi>e</mi> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, we prove that if <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2114_Article_IEq9.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">$m>1,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>></mo> <mn>1</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2114_Article_IEq10.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="TEX">$(s+1) \ge q$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>≥</mo> <mi>q</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2114_Article_IEq11.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="102" /> </InlineMediaObject> <EquationSource Format="TEX">$2n \ge f(q-1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mi>n</mi> <mo>≥</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, then G does not have a normal complement in V(FG).

On the normal complement problem for finite group algebras of abelian-by-cyclic groups

期刊论文
发表日期：

2025年3月23日

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Allen Herman,
Surinder Kaur

摘要

Assume F is a finite field of order $p^f$ $p^{f}$ and q is an odd prime for which $p^f-1=sq^m$ $p^{f} - 1 = s q^{m}$ , where $m \ge 1$ $m \geq 1$ and $(s,q)=1$ $(s, q) = 1$ . In this article, we obtain the order of the symmetric and the unitary subgroup of the semisimple group algebra $FC_q.$ $F C_{q} .$ Further, for the extension G of $C_q = \langle b \rangle $ $C_{q} = ⟨ b ⟩$ by an abelian group A of order $p^n$ $p^{n}$ with $C_{A}(b) = \{e\}$ $C_{A} (b) = {e}$ , we prove that if $m>1,$ $m > 1,$ or $(s+1) \ge q$ $(s + 1) \geq q$ and $2n \ge f(q-1)$ $2 n \geq f (q - 1)$ , then G does not have a normal complement in V(FG).