In this book we analyse unit groups of group algebras KG for non-abelian p-groups G and fields
K of characteristic p. By calculating the core and the normaliser of U in 1 + rad(KG) - the
group of normalized units -- for every subgroup U of G we generalise results of K.R. Pearson
and D.B. Coleman using fixed points of enhanced group actions. Our concept of so-called
end-commutable ordering leads to a new method of studying the center of 1 + rad(KG). We proof
that a finite group G is nilpotent if and only if every conjugacy class possesses an
end-commutable ordering. As a simple consequence we get a result of A.A. Bovdi and Z. Patay
which shows how the exponent of the center of 1 + rad(KG) can be determined by calculations
purely within the group G. We describe the groups for which this exponent is extremal and
calculate the exponent for various group classes (e.g. regular groups special groups Sylow
subgroups of linear and symmetric groups) and group constructions (e.g. wreath products
central products special group extensions isoclinic groups). Another application of our
concept of end-commutable ordering is a description of the invariants of the center of 1 +
rad(KG) for a finite field K. They are determined purely by the group G and the field K and can
be visualized by a special graph - the class-graph. As a consequence of our results we prove
that the center the derived subgroups and the p-th-power subgroup of 1 + rad(KG) are not
cyclic. Furthermore we obtain some properties of unit groups of group algebras for
extra-special 2-groups and fields of characteristic 2. Finally we investigate the behaviour of
the center and other characteristics (e.g. the exponent the class of nilpotency the Baer
length the degree of commutativity) for the chain of iterated unit groups of modular group
algebras. For this we use Lie and radical algebra methods.
