Let G be a finite group and $\mathrm {Irr}(G)$ the set of all irreducible complex characters of G. Define the codegree of $\chi \in \mathrm {Irr}(G)$ as $\mathrm {cod}(\chi ):={|G:\mathrm {ker}(\chi ) |}/{\chi (1)}$ and let $\mathrm {cod}(G):=\{\mathrm {cod}(\chi ) \mid \chi \in \mathrm {Irr}(G)\}$ be the codegree set of G. Let $\mathrm {A}_n$ be an alternating group of degree $n \ge 5$. We show that $\mathrm {A}_n$ is determined up to isomorphism by $\operatorname {cod}(\mathrm {A}_n)$.

The codegree of an irreducible character $\chi $ of a finite group G is $|G : \ker \chi |/\chi (1)$. We show that the Ree group ${}^2G_2(q)$, where $q = 3^{2f+1}$, is determined up to isomorphism by its set of codegrees.

We present a family of counterexamples to a question proposed recently by Moretó concerning the character codegrees and the element orders of a finite solvable group.

Let G be a finite group, let ${\mathrm{Irr}}(G)$ be the set of all irreducible complex characters of G and let $\chi \in {\mathrm{Irr}}(G)$ . Define the codegrees, ${\mathrm{cod}}(\chi ) = |G: {\mathrm{ker}}\chi |/\chi (1)$ and ${\mathrm{cod}}(G) = \{{\mathrm{cod}}(\chi ) \mid \chi \in {\mathrm{Irr}}(G)\} $ . We show that the simple group ${\mathrm{PSL}}(2,q)$ , for a prime power $q>3$ , is uniquely determined by the set of its codegrees.

Let G be a finite group and $\chi $ be a character of G. The codegree of $\chi $ is ${{\operatorname{codeg}}} (\chi ) ={|G: \ker \chi |}/{\chi (1)}$ . We write $\pi (G)$ for the set of prime divisors of $|G|$ , $\pi ({{\operatorname{codeg}}} (\chi ))$ for the set of prime divisors of ${{\operatorname{codeg}}} (\chi )$ and $\sigma ({{\operatorname{codeg}}} (G))= \max \{|\pi ({{\operatorname{codeg}}} (\chi ))| : \chi \in {\textrm {Irr}}(G)\}$ . We show that $|\pi (G)| \leq ({23}/{3}) \sigma ({{\operatorname{codeg}}} (G))$ . This improves the main result of Yang and Qian [‘The analog of Huppert’s conjecture on character codegrees’, J. Algebra478 (2017), 215–219].

Let $G$ be a $p$-group and let $\unicode[STIX]{x1D712}$ be an irreducible character of $G$. The codegree of $\unicode[STIX]{x1D712}$ is given by $|G:\,\text{ker}(\unicode[STIX]{x1D712})|/\unicode[STIX]{x1D712}(1)$. If $G$ is a maximal class $p$-group that is normally monomial or has at most three character degrees, then the codegrees of $G$ are consecutive powers of $p$. If $|G|=p^{n}$ and $G$ has consecutive $p$-power codegrees up to $p^{n-1}$, then the nilpotence class of $G$ is at most 2 or $G$ has maximal class.