A finite group G is called an m-cyclic group if it has exactly m cyclic subgroups (including the identity subgroup). For 2 ⩽ m ⩽ 12, the m-cyclic groups have been classified in a series of papers. We push the above research work further to classify the finite 13-cyclic groups, which could be considered as a step to answer the open problem posed by M. Tărnăuceanu (2015). The detailed structure of many groups of “small” orders is also analyzed. The following main theorem is proved: Let G be a finite 13-cyclic group. Then ∣π(G)∣ ⩽ 2, and one of the following holds:<OrderedList> <ListItem> <ItemNumber>(1)</ItemNumber> <ItemContent> ∣π(G)∣ = 1, G ≅ Q32, Z11 × Z11 or <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10587_2025_4124_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbf{Z}_{p^{11}}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mrow> <mi mathvariant="bold">Z</mi> </mrow> <mrow> <msup> <mi>p</mi> <mrow> <mn>11</mn> </mrow> </msup> </mrow> </msub> </math></EquationSource> </InlineEquation> with p a prime. </ItemContent> </ListItem> <ListItem> <ItemNumber>(2)</ItemNumber> <ItemContent> ∣π(G)∣ = 2, G ≅ D22, SL(2, 3), Z11: Z5, Z7: Z8, Z7: Z27, Z5: Z16, or Z3: Z32. </ItemContent> </ListItem> </OrderedList>

Finite groups with a small number of cyclic subgroups

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发表日期：

2025年4月9日

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Hailin Liu,
Xiangyu Chen,
Shouhong Qiao

摘要

A finite group G is called an m-cyclic group if it has exactly m cyclic subgroups (including the identity subgroup). For 2 ⩽ m ⩽ 12, the m-cyclic groups have been classified in a series of papers. We push the above research work further to classify the finite 13-cyclic groups, which could be considered as a step to answer the open problem posed by M. Tărnăuceanu (2015). The detailed structure of many groups of “small” orders is also analyzed. The following main theorem is proved: Let G be a finite 13-cyclic group. Then ∣π(G)∣ ⩽ 2, and one of the following holds:

(1)

∣π(G)∣ = 1, G ≅ Q₃₂, Z₁₁ × Z₁₁ or $\mathbf{Z}_{p^{11}}$ $Z_{p^{11}}$ with p a prime.

(2)

∣π(G)∣ = 2, G ≅ D₂₂, SL(2, 3), Z₁₁: Z₅, Z₇: Z₈, Z₇: Z₂₇, Z₅: Z₁₆, or Z₃: Z₃₂.