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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 
$\operatorname {cod}(\mathrm {A}_n)$.
$\boldsymbol {{}^{2}G_{2}(q)}$ USING THE CODEGREE SET
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.
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].
$p$-groups
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.
