A new identity in the Lie ring of a free group of prime exponent, and groups without the Hughes property

A multilinear identity of degree $3p-2$ is given in explicit form, and it is shown that this identity holds in the associated Lie ring of a free group of prime exponent $p$. It is also shown that if this identity is not a consequence of the known identities of Wall of degree $2p-1$ and the $(p-1)$st Engel identity, there exists a finite $p$-group in which the index of the (nontrivial) Hughes subgroup is $p^3$.
Bibliography: 13 titles.