6533b7dbfe1ef96bd1270046

RESEARCH PRODUCT

On Join Properties of Hall π-Subgroups of Finite π-Soluble Groups

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All groups considered in the sequel are finite. K. Doerk and T. Hawkes, in Section I.4 of their recent comprehensive w x volume on finite soluble groups 1 , include background material and a proof of the following result: Let S be a Hall system of a soluble group G and let U and V be subgroups into which S reduces. Then S reduces into U l V, and if , in addition, U permutes with V, then S reduces into UV. It is clear that the second part of the above result holds equally well with a single Hall subgroup in place of a Hall system; in other words, if a Hall p-subgroup of G contains Hall p-subgroups of U and V and U permutes with V, then it also contains a Hall p-subgroup of UV.