Let
p
be a prime, and let
Λ
2
p
be a connected cubic arc-transitive graph of order 2
p
. In the literature, a lot of works have been done on the classification of edge-transitive normal covers of
Λ
2
...p
for specific
p
≤
7
. An interesting problem is to generalize these results to an arbitrary prime
p
. In 2014, Zhou and Feng classified edge-transitive cyclic or dihedral normal covers of
Λ
2
p
for each prime
p
. In our previous work, we classified all edge-transitive
N
-normal covers of
Λ
2
p
, where
p
is a prime and
N
is a metacyclic 2-group. In this paper, we give a classification of edge-transitive
N
-normal covers of
Λ
2
p
, where
p
≥
5
is a prime and
N
is a metacyclic group of odd prime power order.
Let
G
be a finite non-Abelian
p
-group, where
p
is an odd prime, such that
G/Z
(
G
) is metacyclic. We prove that all commuting automorphisms of
G
form a subgroup of Aut(
G
) if and only if
G
is of ...nilpotence class 2.
Let G be a finite group. A subset X of G is a set of pairwise noncommuting elements if any two distinct elements of X do not commute. In this paper we determine the maximum size of these subsets in ...any finite nonabelian metacyclic p-group for an odd prime p.
Let ω be a 2-cocycle of a metacyclic p-group G representing a non-trivial element of the Schur multiplier
Then the number of ω-regular conjugacy classes of G, the subgroup consisting of the ω-regular ...elements in the center of G, the degree of each irreducible ω-character of G and a representation group H of G with M(H) trivial are all determined. Finally, for ω constructed from H, the projective character table of G corresponding to ω is found in the case that G is of positive type.
Communicated by Mark Lewis
Given any group G, the normalizer
of the subgroup of left translations in the group of all permutations on G is called the holomorph, and the normalizer
of
in turn is called the multiple holomorph. ...The quotient
has been computed for various families of groups G in the literature. In this paper, we shall supplement the existing results by considering finite split metacyclic p-groups G with p an odd prime. Our work gives new examples of groups G for which T(G) is not a 2-group.
The Amit conjecture about word maps on finite nilpotent groups has been shown to hold for certain classes of groups. The generalised Amit conjecture says that the probability of an element occurring ...in the image of a word map on a finite nilpotent group
G
is either 0, or at least 1/|
G
|. Noting the work of Ashurst, we name the generalised Amit conjecture the Amit–Ashurst conjecture and show that the Amit–Ashurst conjecture holds for finite
p
-groups with a cyclic maximal subgroup.
For every prime
p
≥
5
, we give examples of Beauville
p
-groups whose Beauville structures are never strongly real. This shows that there are nilpotent purely non-strongly real Beauville groups. On ...the other hand, we determine infinitely many Beauville 2-groups which are purely strongly real. This answers two questions formulated by Fairbairn (Arch Math.
https://doi.org/10.1007/s00013-018-1288-4
).
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