Communicated by M. Sapir MSC: Primary: 20D45 Secondary: 17B70; 20D15; 20F40 a b s t r a c t Suppose that a finite group G admits a Frobenius group of automorphisms FH of coprime order with kernel F and complement H such thatis repeated k times, then G is nilpotent of class bounded in terms of c, k and |H| only. It is also proved that if F is abelian of rank at least three and C G (a) is nilpotent of class at most d for every a ∈ F \ {1}, then G is nilpotent of class bounded in terms of c, d and |H|. The proofs are based on results on graded Lie rings.
Let q be a prime and A a finite q-group of exponent q acting by automorphisms on a finite q ′ -group G. Assume that A has order at least q 3 . We show that if γ ∞ (C G (a)) has order at most m for any a ∈ A # , then the order of γ ∞ (G) is bounded solely in terms of m and q. If γ ∞ (C G (a)) has rank at most r for any a ∈ A # , then the rank of γ ∞ (G) is bounded solely in terms of r and q.1991 Mathematics Subject Classification. 20D45.
Let
p
p
be a prime and
A
A
a finite group of exponent
p
p
acting by automorphisms on a finite
p
′
p’
-group
G
G
. Assume that
A
A
has order at least
p
3
p^3
and
C
G
(
a
)
C_G(a)
is nilpotent of class at most
c
c
for any
a
∈
A
#
a\in A^{\#}
. It is shown that
G
G
is nilpotent with class bounded solely in terms of
c
c
and
p
p
.
Let q be a prime and A an elementary abelian group of order at least q 3 acting by automorphisms on a finite q ′ -group G. It is proved that if |γ ∞ (C G (a))| ≤ m for any a ∈ A # , then the order of γ ∞ (G) is m-bounded. If F (C G (a)) has index at most m in C G (a) for any a ∈ A # , then the index of F 2 (G) is m-bounded.1991 Mathematics Subject Classification. 20D45.
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