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Journal of algebraic combinatorics, 2024-01, Vol.59 (1), p.23-35
Ort / Verlag
New York: Springer US
Erscheinungsjahr
2024
Quelle
SpringerLink
Beschreibungen/Notizen
Let
M
be an orientably regular (resp. regular) map with the number
n
vertices. By
G
+
(resp.
G
) we denote the group of all orientation-preserving automorphisms (resp. all automorphisms) of
M
. Let
π
be the set of prime divisors of
n
. A Hall
π
-subgroup of
G
+
(resp.
G
) is meant a subgroup such that the prime divisors of its order all lie in
π
and the primes of its index all lie outside
π
. It is mainly proved in this paper that (1) suppose that
M
is an orientably regular map where
n
is odd. Then
G
+
is solvable and contains a normal Hall
π
-subgroup; (2) suppose that
M
is a regular map where
n
is odd. Then
G
is solvable if it has no composition factors isomorphic to
PSL
(
2
,
q
)
for any odd prime power
q
≠
3
, and
G
contains a normal Hall
π
-subgroup if and only if it has a normal Hall subgroup of odd order.