We show that elementary abelian direct factors can be disregarded in the study of the modular isomorphism problem. Moreover, we obtain four new series of abelian invariants of the group base in the ...modular group algebra of a finite p-group. Finally, we apply our results to new classes of groups.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
Let
p
be an odd prime number. We show that the modular isomorphism problem has a positive answer for finite
p
-groups whose center has index
p
3
, which is a strong contrast to the analogous ...situation for
p
=
2
.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
Let
p
be a prime and let
G
be a finite
p
-group. We show that the isomorphism type of the maximal abelian direct factor of
G
, as well as the isomorphism type of the group algebra over
F
p
of the ...non-abelian remaining direct factor, if existing, are determined by
F
p
G
, generalizing the main result in Margolis et al. (Abelian invariants and a reduction theorem for the modular isomorphism problem, Journal of Algebra
636
, 533-559 (2023)) over the prime field. To do this, we address the problem of finding characteristic subgroups of
G
such that their relative augmentation ideals depend only on the
k
-algebra structure of
kG
, where
k
is any field of characteristic
p
, and relate it to the modular isomorphism problem, extending and reproving some known results.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
Let
p
be a an odd prime and let
G
be a finite
p
-group with cyclic commutator subgroup
G
′
. We prove that the exponent and the abelianization of the centralizer of
G
′
in
G
are determined by the ...group algebra of
G
over any field of characteristic
p
. If, additionally,
G
is 2-generated then almost all the numerical invariants determining
G
up to isomorphism are determined by the same group algebras; as a consequence the isomorphism type of the centralizer of
G
′
is determined. These claims are known to be false for
p
= 2.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
Let
KG
be the group algebra of a group
G
over a field
K
of characteristic
p
>
0
. The classification of group algebras
KG
for which Lie nilpotency indices are maximal or almost maximal has already ...been determined. After that Bovdi and Srivastava classified Lie nilpotent group algebras with
t
L
(
K
G
)
=
|
G
′
|
-
2
p
+
3
,
|
G
′
|
-
3
p
+
4
and
|
G
′
|
-
4
p
+
5
.
In this paper, our aim is to classify the group algebras
KG
for which
t
L
(
K
G
)
=
|
G
′
|
-
5
p
+
6
,
|
G
′
|
-
6
p
+
7
and
|
G
′
|
-
7
p
+
8
.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OBVAL, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
We give a full description of locally finite 2-groups
G
such that the normalized group of units of the group algebra
FG
over a field
F
of characteristic 2 has exponent 4.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OBVAL, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
A p-group is called powerful if every commutator is a product of pth powers when p is odd and a product of fourth powers when p=2. In the group algebra of a group G of p-power order over a finite ...field of characteristic p, the group of normalized units is always a p-group. We prove that it is never powerful except, of course, when G is abelian.
It is shown that the isomorphism type of a metacyclic p-group is determined by its group algebra over the field F of p elements. This completes work of Baginski. It is also shown that, if a p-group G ...has a cyclic commutator subgroup G', then the order of the largest cyclic subgroup containing G' is determined by FG.
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BFBNIB, INZLJ, NMLJ, NUK, PNG, SAZU, UL, UM, UPUK, ZRSKP
We show that p-groups of maximal class and order p
5 are determined by their group algebras over the field of p elements. The most important information requisite for the proof is obtained from a ...detailed study of the unit group of a quotient algebra of the group algebra, larger than the small group algebra.
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