Abstract
AbstractA skew morphism of a finite group A is a permutation $$\varphi $$
φ
of A fixing the identity element and for which there is an integer-valued function $$\pi $$
π
on A such that $$\varphi (ab)=\varphi (a)\varphi ^{\pi (a)}(b)$$
φ
(
a
b
)
=
φ
(
a
)
φ
π
(
a
)
(
b
)
for all $$a, b \in A$$
a
,
b
∈
A
. A skew morphism $$\varphi $$
φ
of A is smooth if the associated power function $$\pi $$
π
is constant on the orbits of $$\varphi $$
φ
, that is, $$\pi (\varphi (a))\equiv \pi (a)\pmod {|\varphi |}$$
π
(
φ
(
a
)
)
≡
π
(
a
)
(
mod
|
φ
|
)
for all $$a\in A$$
a
∈
A
. In this paper, we show that every skew morphism of a cyclic group of order n is smooth if and only if $$n=2^en_1$$
n
=
2
e
n
1
, where $$0 \le e \le 4$$
0
≤
e
≤
4
and $$n_1$$
n
1
is an odd square-free number. A partial solution to a similar problem on non-cyclic abelian groups is also given.
Funder
National Research Foundation of Korea
Javna Agencija za Raziskovalno Dejavnost RS
Publisher
Springer Science and Business Media LLC
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