2022/01/22 by Benjamin Steinberg, Steinberg, Benjamin
Mathematics · #20C15 #20M30 #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2201.09129
openalex publication_date 2022/01/22 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28
Gaschütz (1954) proved that a finite group G has a faithful irreducible complex representation if and only if its socle is generated by a single element as a normal subgroup; this result extends to arbitrary fields of characteristic p so long as G has no nontrivial normal p-subgroup. Žmud' (1956) showed that the minimum number of irreducible constituents in a faithful complex representation of G coincides with the minimum number of generators of its socle as a normal subgroup; this result can also be extended to arbitrary fields of any characteristic p such that G has no nontrivial normal p-subgroup (i.e., over which G admits a faithful completely reducible representation). Rhodes (1969) characterized the finite semigroups admitting a faithful irreducible representation over an arbitrary field as generalized group mapping semigroups over a group admitting a faithful irreducible representation over the field in question. Here, we provide a common generalization of the theorems of Žmud' and Rhodes by determining the minimum number of irreducible constituents in a faithful completely reducible representation of a finite semigroup over an arbitrary field (provided that it has one). Our key tool for the semigroup result is a relativized version of Žmud''s theorem that determines, given a finite group G and a normal subgroup N\lhd G, what is the minimum number of irreducible constituents in a completely reducible representation of G whose restriction to N is faithful.