Biggs tree groups

2025/11/17 by Christopher H. Cashen, Cashen, Christopher H.
Computer Science · Mathematics · #05C38 #20B20 #Advanced Combinatorial Mathematics #Advanced Graph Theory Research #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Primary 20B05 #Secondary 05C25

paper · pdf · doi:10.48550/arxiv.2511.13292

openalex publication_date 2025/11/17 · openalex created_date 2025/11/19 · openalex updated_date 2026/07/28

Abstract

Biggs gave an explicit construction, using finite colored trees, of finite permutation groups whose Cayley graphs have valence \(C\) and girth tending to infinity as the radius \(R\) of the tree tends to infinity. We show that when the number of colors is at least 3, the group so presented contains the full alternating group on the vertices of the tree. This gives, for each \(C≥ 3\), an infinite family of pairs \((G_C,R,S_C,R)\) such that \(G_C,R\) is an alternating or symmetric group, \(S_C,R\) is a generating set of \(G_C,R\) of size \(C\) with an explicit permutation description of its generators, and such that the sequence of Cayley graphs \(Cay(G_C,R,S_C,R)\) has constant valence \(C\) and girth tending to infinity as \(R\) tends to infinity.

Citations

• Acyclicity in finite groups and groupoids
• On the maximum orders of elements of finite almost simple groups and\n primitive permutation groups
• Symmetric Groups and Expander Graphs

Related

• Symmetric Groups and Expanders
• Symmetric Groups and Expander Graphs
• Finite Groups of Uniform Logarithmic Diameter
• The invariably generating graph of the alternating and symmetric groups
• Cayley graphs formed by conjugate generating sets of S_n