A Noncoprime Shult Type Theorem.
In this note we study finite -groups admitting a factorization by an Abelian subgroup and a subgroup . As a consequence of our results we prove that if contains an Abelian subgroup of index then has derived length at most .
We prove that pure subgroups of thick Abelian -groups which are modulo countable are again thick. This generalizes a result due to Megibben (Michigan Math. J. 1966). Some related results are also established.