Let
f
:
T
→
R
be of class
C
1
+
δ
for some
δ
>
0
and let
c
∈
Z
. We show that for a generic
α
∈
R
, the extension
T
c
,
f
:
T
2
→
T
2
of the irrational rotation
T
x
=
x
+
α
, given by
T
c
,
f
(
x
,
u
...)
=
(
x
+
α
,
u
+
c
x
+
f
(
x
)
)
(
mod
1
) satisfies Sarnak’s conjecture.
We investigate the ultrasolvability problem for minimal p-group extensions of odd order: for the factorgroup of such extension, there exists a Galois extension of number fields such as corresponding ...embedding problem is ultrasolvable (i.e. this embedding problem is solvable and all its solutions are fields).
The Suzuki simple group
Sz
(8) has an automorphism group 3. Using the electronic Atlas
22
, the group
Sz
(8) : 3 has an absolutely irreducible module of dimension 12 over
F
2
.
Therefore a split ...extension group of the form
2
12
:
(
S
z
(
8
)
:
3
)
:
=
G
¯
exists. In this paper we study this group, where we determine its conjugacy classes and character table using the coset analysis technique together with Clifford-Fischer Theory. We determined the inertia factor groups of
G
¯
by analysing the maximal subgroups of
Sz
(8) : 3 and maximal of the maximal subgroups of
Sz
(8) : 3 together with various other information. It turns out that the character table of
G
¯
is a
43
×
43
complex valued matrix, while the Fischer matrices are all integer valued matrices with sizes ranging from 1 to 7.
In this paper we use the Galois module structure for the classical parameterizing spaces for elementary p-abelian extensions of a field K to give necessary and sufficient conditions for the ...solvability of any embedding problem which is an extension of Z/pnZ with elementary p-abelian kernel. This allows us to count the total number of solutions to a given embedding problem when the appropriate modules are finite, and leads to some nontrivial automatic realization and realization multiplicity results for Galois groups.
Determining the length of short conjugators in a group can be considered as an effective version of the conjugacy problem. The conjugacy length function provides a measure for these lengths. We study ...the behavior of conjugacy length functions under group extensions, introducing the twisted and restricted conjugacy length functions. We apply these results to show that certain abelian-by-cyclic groups have linear conjugacy length function and certain semidirect products ℤ
d
⋊ ℤ
k
have at most exponential (if k > 1) or linear (if k = 1) conjugacy length functions.
The simple symplectic group
Sp
(8, 2) has 11 conugacy classes of maximal subgroups. The fourth maximal subgroup of
Sp
(8, 2) is a group of the form
2
10
:
A
8
:
=
G
¯
.
In this paper we study this ...group, where we determine its conjugacy classes and character table using the coset analysis technique together with Clifford–Fischer Theory. We determined the inertia factor groups of
G
¯
and there are 7 such groups having the forms:
H
1
=
A
8
,
H
2
=
2
3
:
G
L
(
3
,
2
)
,
H
3
=
2
4
:
(
S
3
×
S
3
)
,
H
4
=
2
3
:
S
4
,
H
5
=
S
5
,
H
6
=
(
S
3
×
S
3
)
:
2
and
H
7
=
2
×
S
4
.
The character table of
G
¯
is a
81
×
81
complex valued matrix, while the Fischer matrices are all integer valued matrices with sizes ranging from 1 to 16.
On a group of the form 214:Sp(6, 2) Basheer, Ayoub B.M.; Seretlo, Thekiso T.
Quaestiones mathematicae,
2/15/2016, Letnik:
39, Številka:
1
Journal Article
Recenzirano
The symplectic group Sp(6, 2) has a 14−dimensional absolutely irreducible module over
. Hence a split extension group of the form Ḡ = 2
14
:Sp(6, 2) does exist. In this paper we first determine the ...conjugacy classes of Ḡ using the coset analysis technique. The structures of inertia factor groups were determined. The inertia factor groups are Sp(6, 2), (2
1+4
× 2
2
):(S
3
× S
3
), S
3
× S
6
, PSL(2, 8), (((2
2
×Q
8
):3):2):2, S
3
×A
5
,and 2×S
4
×S
3
.We then determine the Fischer matrices and apply the Clifford-Fischer theory to compute the ordinary character table of
. The Fischer matrices of
are all integer valued, with size ranging from 4 to 16. The full character table of
is a 186 × 186 complex valued matrix.
The purpose of this paper is the determination of the inertia factors, the computations of the Fischer matrices and the ordinary character table of the split extension ...$overline{G}= 3^{7}{:}Sp(6,2)$ by means of Clifford-Fischer Theory. We firstlydetermine the conjugacy classes of $overline{G}$ using the coset analysis method. The determination of the inertia factor groups of this extension involved looking at some maximal subgroups of the maximal subgroups of $Sp(6,2).$ The Fischer matrices of $overline{G}$ are all listed in this paper and their sizes range between 2 and 10. The character table of $overline{G},$ which is a $118times 118 mathbb{C}$-valued matrix, is available in the PhD thesis of the first author, which could be accessed online.
We study the behaviour of endomorphisms and automorphisms of groups involved in abelian group extensions. The main result can be stated as follows: Let 0→N→G→Q→1 be an abelian group extension. Then ...one has the following exact sequence of groups:
where End
N,Q
(G) denotes the set of all endomorphisms of G which centralise N and induce identity on Q,
denotes the set of all endomorphisms of G which normalise N and induce identity on Q and End
Q
(N) denotes the set of endomorphisms of N which are compatible with the action of Q on N. This exact sequence is obtained using the five-term exact sequence in group cohomology.
An interesting fact we discovered is that the first three terms involved have ring structure and the maps between them are ring homomorphisms. The ring structure on End
Q
(N) is well-known, however the ring structure of the second term is a little more exotic. Restricted on quasi-regular elements, this gives the exact sequence proved recently in
4
by Passi, Singh and Yadav.
PDF
