Given a finite group $G$, define the minimal degree $\unicode[STIX]{x1D707}(G)$ of $G$ to be the least $n$ such that $G$ embeds into $S_{n}$. We call $G$ exceptional if there is some $N\unlhd G$ with $\unicode[STIX]{x1D707}(G/N)>\unicode[STIX]{x1D707}(G)$, in which case we call $N$ distinguished. We prove here that a subgroup with no abelian composition factors is not distinguished.

For a finite group $G$, denote by $\unicode[STIX]{x1D707}(G)$ the degree of a minimal permutation representation of $G$. We call $G$ exceptional if there is a normal subgroup $N\unlhd G$ with $\unicode[STIX]{x1D707}(G/N)>\unicode[STIX]{x1D707}(G)$. To complete the work of Easdown and Praeger [‘On minimal faithful permutation representations of finite groups’, Bull. Aust. Math. Soc.38(2) (1988), 207–220], for all primes $p\geq 3$, we describe an exceptional group of order $p^{5}$ and prove that no exceptional group of order $p^{4}$ exists.

For a finite group G, we denote by μ(G) the minimum degree of a faithful permutation representation of G. We prove that if G is a finite p-group with an abelian maximal subgroup, then μ(G/G′)≤μ(G).

In this note we show that the image of the transfer for permutation representations of finite groups is generated by the transfers of special monomials. This leads to a description of the image of the transfer of the alternating groups. We also determine the height of these ideals.