Maximal representation dimension of finitep-groups

Abstract

Abstract. The representation dimension rdim(G) of a finite group G is the smallest positive integer m for which there exists an embedding

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“…Suppose χ is a faithful character of G. From [1, Lemma 3.5], it follows that χ is a sum of two irreducible characters of G. Consequently, the degree of χ is either p + 1 or 2p. However, in view of Lemma 15 of [9], if p is an odd prime, then the deg(χ) ≤ p + 1; therefore, deg(χ) = p + 1. For an arbitrary group G of order 16, one can verify from the character table of G that δ(G) = p + 1.…”

“…is the subgroup generated by elements g ∈ Z(G) such that g p = 1. From Lemma 14 of[9], it follows that δ(G) ≤ p + 1. As Z(G) is non cyclic, the group G does not have an irreducible faithful character.…”

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“…For finite groups faith(S) is a well-known invariant studied since the early days of representation theory. It is sometimes called representation dimension, and has attracted recent attention, see [Mo21] and the references therein, including [CKR11] or [BMKS16]. Various versions of faithfulness have been studied in monoid theory as well, see for example [MS12b] who call the faithfulness the effective dimension.…”