1,720,976 research outputs found
On the number of fixed edges of automorphisms of vertex-transitive graphs of small valency
Full text linkWe prove that, if Γ is a finite connected 3-valent vertex-transitive, or 4-valent vertex- and edge-transitive graph, then either Γ is part of a well-understood family of graphs, or every non-identity automorphism of Γ fixes at most 1/3 of the edges. This answers a question proposed by Primož Potočnik and the third author
On the order of semiregular automorphisms of cubic vertex-transitive graphs
Full text linkWe prove that, if P is a finite connected cubic vertex-transitive graph, then either there exists a semiregular automorphism of P of order at least 6, or the number of vertices of P is bounded above by an absolute constant. (c) 2024 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/)
FINITE GROUPS WITH INDEPENDENT GENERATING SETS OF ONLY TWO SIZES
Full text linkA generating set S for a group G is independent if the subgroup generated by is properly contained in G for all We describe the structure of finite groups G such that there are precisely two numbers appearing as the cardinalities of independent generating sets for G
Hypermaps Over Non-Abelian Simple Groups and Strongly Symmetric Generating Sets
Full text linkA generating pair x, y for a group G is said to be symmetric if there exists an automorphism φx,y of G inverting both x and y, that is, xφx,y = x−1 and yφx,y = y−1. Similarly, a group G is said to be strongly symmetric if G can be generated with two elements and if all generating pairs of G are symmetric. In this paper we classify the finite strongly symmetric non-abelian simple groups. Combinatorially, these are the finite non-abelian simple groups G such that every orientably regular hypermap with monodromy group G is reflexible. Mathematics Subject Classifications: 05C10, 05C25, 20B25
Independent sets of generators of prime power order
Full text linkA subset X of a finite group G is said to be prime-power-independent if each element in X has prime power order and there is no proper subset Y of X with 〈Y,Φ(G)〉=〈X,Φ(G)〉, where Φ(G) is the Frattini subgroup of G. A group G is Bpp if all prime-power-independent generating sets for G have the same cardinality. We prove that, if G is Bpp, then G is solvable. Pivoting on some recent results of Krempa and Stocka (2014); Stocka (2020), this yields a complete classification of Bpp-groups
The Engel graph of almost simple groups
No full textThe Engel graph of a finite group G, denoted Gamma(G), is a directed graph encoding the pairs of elements in G satisfying some Engel word. Recent work of Detomi, Lucchini and Nemmi shows that Gamma(G) is connected and the problem of understanding the strong connectivity of this graph has been reduced to the case where G is an almost simple group. In this paper, we complete the analysis by determining when Gamma(G) is strongly connected in the almost simple setting; the only exceptions involve groups with socle PSL2(q) or 2B2(q)
On the diameter of Engel graphs
No full textGiven a finite group G, the Engel graph of G is a directed graph Gamma(G) encoding pairs of elements satisfying some Engel word. Namely, Gamma(G) is the directed graph, where the vertices are the non-hypercentral elements of G and where there is an arc from x to y if and only if [x,(n)y]=1 for some n is an element of N. From previous work, it is known that, except for a few exceptions, Gamma(G) is strongly connected. In this paper, we give an absolute upper bound on the diameter of Gamma(G), when Gamma(G) is strongly connected
Milnor-Wolf Theorem for group endomorphisms
Full text linkWe study the growth of group endomorphisms and we prove an analogue of Chou’s extension of Milnor-Wolf Theorem. Indeed, if G is an elementary amenable group and φ:G→G is an endomorphism, then φ has either polynomial or exponential growth. This result follows by studying the growth of automorphisms of finitely generated groups, where we prove some stronger results
ON THE CAYLEYNESS OF PRAEGER–XU GRAPHS
Full text linkWe give a sufficient and necessary condition for a Praeger–Xu graph to be a Cayley graph
A subexponential bound on the cardinality of abelian quotients in finite transitive groups
Full text linkWe show that, for every transitive permutation group (Formula presented.) of degree (Formula presented.), the largest abelian quotient of (Formula presented.) has cardinality at most (Formula presented.). This gives a positive answer to a 1989 outstanding question of László Kovács and Cheryl Praeger
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