Let D be a division ring with centre F. Assume that M is a maximal subgroup of GLn(D) (n ≥ 1) such that Z(M) is algebraic over F. Group identities on M and polynomial identities on the F-linear hull F[M] are investigated. It is shown that if F[M]is a PI-algebra, then [D: F] ＜∞. When D is non-commutative and F is infinite, it is also proved that if M satisfies a group identity and F[M] is algebraic over F, then we have either M = K* where K is a field and [D: F] ＜∞, or M is absolutely irreducible. For a finite dimensional division algebra D, assume that N is a subnormal subgroup of GLn(D)and M is a maximal subgroup of N. If M satisfies a group identity, it is shown that M is abelian-by-finite.