Skew morphisms provide a fundamental tool for the study of regular Cayley maps, and more generally, for finite groups with a group factorization <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$X=GC$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>=</mo> <mi>G</mi> <mi>C</mi> </mrow> </math></EquationSource> </InlineEquation>, where C is cyclic and core-free in X. Furthermore, if <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$G\cap C=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>∩</mo> <mi>C</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, then <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$X=GC$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>=</mo> <mi>G</mi> <mi>C</mi> </mrow> </math></EquationSource> </InlineEquation> is called the skew product group associated with G and C. In this paper, we investigate the skew product groups of abelian groups <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$G\cong Z_{2^{n-1}}\times Z_2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>≅</mo> <msub> <mi>Z</mi> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </msub> <mo>×</mo> <msub> <mi>Z</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>. First, we show that either <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$X\cong A_4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>≅</mo> <msub> <mi>A</mi> <mn>4</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>, or X is a 2-group, and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$G_X\ne 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>G</mi> <mi>X</mi> </msub> <mo>≠</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Furthermore, we classify the skew product groups of G such that <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$G_X=G$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>G</mi> <mi>X</mi> </msub> <mo>=</mo> <mi>G</mi> </mrow> </math></EquationSource> </InlineEquation>, or <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$G_X$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>G</mi> <mi>X</mi> </msub> </math></EquationSource> </InlineEquation> is a maximal subgroup of G.

On the skew product groups of $Z_{2^{n-1}}\times Z_2$

期刊论文
发表日期：

2026年2月18日

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Youxin Li,
Wei Meng,
Jiakuan Lu

摘要

Skew morphisms provide a fundamental tool for the study of regular Cayley maps, and more generally, for finite groups with a group factorization $X=GC$ $X = G C$ , where C is cyclic and core-free in X. Furthermore, if $G\cap C=1$ $G \cap C = 1$ , then $X=GC$ $X = G C$ is called the skew product group associated with G and C. In this paper, we investigate the skew product groups of abelian groups $G\cong Z_{2^{n-1}}\times Z_2$ $G ≅ Z_{2^{n - 1}} \times Z_{2}$ . First, we show that either $X\cong A_4$ $X ≅ A_{4}$ , or X is a 2-group, and $G_X\ne 1$ $G_{X} \neq 1$ . Furthermore, we classify the skew product groups of G such that $G_X=G$ $G_{X} = G$ , or $G_X$ $G_{X}$ is a maximal subgroup of G.

On the skew product groups of \(Z_{2^{n-1}}\times Z_2\)

摘要