1. Normality of quartic cayley graphs on regular ▫$p$▫-groups : a CFSG-free approachKlavdija Kutnar, Aleksander Malnič, Dragan Marušič, 2026, original scientific article Abstract: Relying on the Classification of Finite Simple Groups it was shown by Feng and Xu (Discrete Math., 2005) that every quartic Cayley graph of a regular $p$-group, $p \neq 2,5$, is normal. In this paper a CFSG-free proof of a special case of Feng-Xu theorem is given. Along the way it is also proved that for an arbitrary $p$-group $G$ with a minimum set $\{a,b\}$ of two generators, one of which is of order $p$, in the corresponding Cayley graph $\mathrm{Cay}(G,\{a,a^{-1},b,b^{-1}\})$ the induced action of vertex stabilizer on the neighbors' set is contained in the dihedral group $D_8$.
Keywords: classification of finite simple groups, regular p-group, p-groups of exponent p, Cayley graph, normal graph, automorphism group
Published in RUP: 23.09.2026; Views: 261; Downloads: 5
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2. On 2-integral Cayley graphsAlireza Abdollahi, Majid Arezoomand, Tao Feng, Shixin Wang, 2026, original scientific article Abstract: In this paper, we introduce the concept of k-integral graphs. A graph Γ is called k-integral if the extension degree of the splitting field of the characteristic polynomial of Γ over rational field ℚ is equal to k. We prove that for any positive integers k and Δ, the set of all finite connected graphs with algebraic degree at most k and maximum degree at most Δ is finite. We study 2-integral Cayley graphs over finite groups G with respect to Cayley sets which are a union of conjugacy classes of G. Among other general results, we completely characterize all finite abelian groups having a connected 2-integral Cayley graph with valency 2, 3, 4 and 5. Furthermore, we classify the finite groups G that al Cayley graphs over G with bounded valency are 2-integral.
Keywords: Cayley graph, algebraic degree, characters of groups, integral eigenvalue
Published in RUP: 18.08.2026; Views: 462; Downloads: 7
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4. Some new compatible groupsZhaochen Ding, Gabriel Verret, 2026, original scientific article Abstract: Two finite groups $L_1$ and $L_2$ are compatible if there exists a finite group $G$ with isomorphic normal subgroups $N_1$ and $N_2$ such that $L_1 \cong G/N_1$ and $L_2 \cong G/N_2$. We prove a new sufficient condition for two groups to be compatible. As a corollary, we obtain that nilpotent groups of the same order are compatible, and so are groups of the same square-free order.
Keywords: compatibility, finite nilpotent groups, p-groups
Published in RUP: 20.03.2026; Views: 867; Downloads: 13
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6. On simple groups that admit a string C-group representationDimitri Leemans, Adrien Vandenschrick, 2026, original scientific article Abstract: We prove that a simple group admits at least one string C-group representation if and only if it is not one of PSL(3, q), PSU(3, q),
PSL(4, 2^n), PSU(4, 2^n), PSU(4, 3), PSU(5, 2), A₆, A₇, M₁₁, M₂₂, M₂₃ or McL.
Keywords: simple groups, string C-groups
Published in RUP: 05.01.2026; Views: 914; Downloads: 14
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7. Groups with elements of order 8 do not have the DCI propertyTed Dobson, Joy Morris, Pablo Spiga, 2025, original scientific article Abstract: Let k be odd, and n an odd multiple of 3. Although this can also be deduced from known results, we provide a new proof that Ck ⋊ C₈ and (Cn × C₃) ⋊ C₈ do not have the Directed Cayley Isomorphism (DCI) property. When k is prime, Ck ⋊ C₈ had previously been proved to have the Cayley Isomorphism (CI) property. To the best of our knowledge, the groups Cp ⋊ C₈ (where p is an odd prime) are only the second known infinite family of groups that have the CI property but do not have the DCI property. This also provides a new proof of the result (which follows from known results but was not explicitly published) that no group with an element of order 8 has the DCI property.
One piece of our proof is a new result that may prove to be of independent interest: we show that if a permutation group has a regular subgroup of index 2 then it must be 2-closed.
Keywords: CI property, DCI property, Cayley graphs, Cayley digraphs, 2-closed groups, 2-closure
Published in RUP: 03.11.2025; Views: 930; Downloads: 7
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8. On the wreath product of signed and gain graphs and its spectrumMatteo Cavaleri, Alfredo Donno, Stefano Spessato, 2025, original scientific article Abstract: We introduce a notion of wreath product of two gain graphs (Γ_1, ψ_1, G_1) and (Γ_2, ψ_2, G_2), producing a gain graph over the direct product group G_2|V_Γ1| × G_1, whose underlying graph is the classical wreath product of graphs Γ_1≀Γ_2. By composition with a suitable group homomorphism, our construction produces a signed graph when the two factors are signed graphs. We prove that the wreath product is stable under switching isomorphism. By using group representations, we are able to perform spectral computations on the wreath product: in particular, we determine its largest and its smallest eigenvalue, and we give a description of the spectrum when the first factor is a complex unit complete balanced or antibalanced gain graph, and the second factor is circulant. Finally, when G_1 is a group of permutations of the vertex set of the first factor, and the group G_2 is abelian, we give an alternative definition producing a gain graph over the group wreath product G_1≀G_2, which turns out to be stable under switching equivalence of the second factor, when the first factor is balanced.
Keywords: gain graph, signed graph, wreath product of graphs, wreath product of groups, circulant gain graph, mixed Kronecker product, π-spectrum
Published in RUP: 22.10.2025; Views: 1509; Downloads: 13
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10. Regular and semi-regular representations of groups by posetsJonathan A. Barmak, 2025, original scientific article Abstract: By a result of Babai, with finitely many exceptions, every group G admits a semi-regular poset representation with three orbits, that is, a poset P with automorphism group Aut(P) ≃ G such that the action of Aut(P) on the underlying set is free and with three orbits. Among finite groups, only the trivial group and ℤ_2 have a regular poset representation (i.e. semi-regular with one orbit), however many infinite groups admit such a representation. In this paper we study non-necessarily finite groups which have a regular representation or a semi-regular representation with two orbits. We prove that if G admits a Cayley graph which is locally the Cayley graph of a free group, then it has a semi-regular representation of height 1 with two orbits. In this case we will see that any extension of the integers by G admits a regular representation. Applications are given to finite simple groups, hyperbolic groups, random groups and indicable groups.
Keywords: automorphism group of posets, Cayley graph, Dehn presentation, simple groups, random groups
Published in RUP: 21.10.2025; Views: 1145; Downloads: 10
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