Nilpotent residual and Fitting subgroup of fixed points in finite groups

Abstract: 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

g a m m a_{i n f t y} (C_{G} (a))

has order at most

m

for any

, then the order of

g a m m a_{i n f t y} (G)

is bounded solely in terms of

m

. If the Fitting subgroup of

C_{G} (a)

has index at most

m

for any

, then the second Fitting subgroup of

G

has index bounded solely in terms of

m

.

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