Symmetry And Group

Download Locally Finite Groups by Otto H Kegel, Bertram A F Wehrfritz PDF

By Otto H Kegel, Bertram A F Wehrfritz

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Example text

Clearly, the element z of the (generalized) quaternion group QZn is in the centre of Q2.. One easily shows that the element z is the only involution of the (generalized) quaternion Q,.. 1 one shows that the (generalized) quaternion group Q,, contains a normal cyclic subgroup of index 2, which is characteristic if n > I, and every element of Q",C \ is of order four and inverts every element of C. The group G is called locally quaternion if every finite subset of G is contained in some (generalized) quaternion subgroup of G.

Hartley [3] and [8] takes this further by considering 3-normalizers and 3-abnormal subroups of U-groups. Apart from the counter examples mentioned above there are some positive results which indicate that to insist that the maximalp-subgroups of a group be conjugate, is to impose a severe restriction on that group. For example the class U is much smaller than might at first be apparent and its previously studied subclasses mentioned above are really quite reflective of its complex* This can be weakened somewhat, see B.

Very sharp information has been obtained in Gorenstein and Walter [2] on the structure of finite groups with dihedral Sylow 2-subgroups. 19). Brauer and Suzuki [I] show that a finite group cannot be non-abelian and simple if its Sylow 2-subgroups are (generalized) quaternion. This result has since been considerably extended by Glauberman [l].

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