Frattini subgroups ofp-closed andp-supersolvable groups |
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Authors: | W. Mack Hill |
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Affiliation: | (1) Worcester State College, 01602 Worcester, Massachusetts, U.S.A. |
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Abstract: | A nonabelianp-group with cyclic center cannot occur as a normal subgroup contained in the Frattini subgroup of ap-closed group. If a nonabelian normal subgroup of orderp n and nilpotence classk is contained in the Frattini subgroup of ap-closed group, then its exponent is a divisor ofp n−k . This fact is used to derive a relation among the order, number of generators, exponent, and class of the Frattini subgroup, forp-groups. Finally, it is conjectured that a nonabelianp-group having center of orderp cannot occur as a normal subgroup contained in the Frattini subgroup of any finite group. A proof is given forp-supersolvable groups. |
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