Factorization number and subgroup commutativity degree via spectral invariants
The factorization number F 2 ( G ) of a finite group G is the number of all possible factorizations of G = H K as a product of its subgroups H and K , while the subgroup commutativity degree sd ( G ) of G is the probability of finding two commuting subgroups in G at random. It is known that sd ( G )...
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| Published in | Computational & applied mathematics Vol. 42; no. 3 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Cham
Springer International Publishing
01.04.2023
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
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| Summary: | The factorization number
F
2
(
G
)
of a finite group
G
is the number of all possible factorizations of
G
=
H
K
as a product of its subgroups
H
and
K
, while the subgroup commutativity degree
sd
(
G
)
of
G
is the probability of finding two commuting subgroups in
G
at random. It is known that
sd
(
G
)
can be expressed in terms of
F
2
(
G
)
. Denoting by
L
(
G
)
the subgroups lattice of
G
, the non–permutability graph of subgroups
Γ
L
(
G
)
of
G
is the graph with vertices in
L
(
G
)
\
C
L
(
G
)
(
L
(
G
)
)
, where
C
L
(
G
)
(
L
(
G
)
)
is the smallest sublattice of
L
(
G
)
containing all permutable subgroups of
G
, and edges obtained by joining two vertices
X
,
Y
such that
X
Y
≠
Y
X
. The spectral properties of
Γ
L
(
G
)
have been recently investigated in connection with
F
2
(
G
)
and
sd
(
G
)
. Here we show a new combinatorial formula, which allows us to express
F
2
(
G
)
, and so
sd
(
G
)
, in terms of adjacency and Laplacian matrices of
Γ
L
(
G
)
. |
|---|---|
| ISSN: | 2238-3603 1807-0302 |
| DOI: | 10.1007/s40314-023-02270-5 |