A criterion for nilpotency in finite groups
For a positive integer n , we denote by π ( n ) the set of prime divisors of n . For a group G and a ∈ G , we denote by o ( a ) the order of the element a . We prove that a finite group G is nilpotent if and only if π ( o ( a b ) ) = π ( o ( a ) o ( b ) ) for all a , b ∈ G of coprime orders.
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| Veröffentlicht in: | Archiv der Mathematik 2024, Vol.122 (1), p.13-16 |
|---|---|
| Hauptverfasser: | , , , |
| Format: | Artikel |
| Sprache: | eng |
| Schlagworte: | |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | For a positive integer
n
, we denote by
π
(
n
)
the set of prime divisors of
n
. For a group
G
and
a
∈
G
, we denote by
o
(
a
) the order of the element
a
. We prove that a finite group
G
is nilpotent if and only if
π
(
o
(
a
b
)
)
=
π
(
o
(
a
)
o
(
b
)
)
for all
a
,
b
∈
G
of coprime orders. |
|---|---|
| ISSN: | 0003-889X 1420-8938 |
| DOI: | 10.1007/s00013-023-01925-3 |