Abstract
For a finite group [Formula: see text] denote by [Formula: see text] the genus of the subgroup graph of [Formula: see text] We prove that [Formula: see text] tends to infinity as either the rank of [Formula: see text] or the number of prime divisors of [Formula: see text] tends to infinity.
Highlights
We prove that γ(L(G)) tends to infinity as either the rank of G or the number of prime divisors of |G| tends to infinity
The subgroup graph L(G) of a finite group G is the graph whose vertices are the subgroups of the group and two vertices, H1 and H2, are connected by an edge if and only if H1 ≤ H2 and there is no subgroup K such that H1 ≤ K ≤ H2
A graph is said to be embedded in a surface S when it is drawn on S so that no two edges intersect
Summary
The subgroup graph L(G) of a finite group G is the graph whose vertices are the subgroups of the group and two vertices, H1 and H2, are connected by an edge if and only if H1 ≤ H2 and there is no subgroup K such that H1 ≤ K ≤ H2. We prove that γ(L(G)) tends to infinity as either the rank of G or the number of prime divisors of |G| tends to infinity. Order divisible by at most three different primes and their Sylow subgroups have rank at most 2 (recall that the rank of a finite group G is the minimal number r such that every subgroup of G can be generated by r elements).
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