1 Introduction
Following Jajcay and Širáň [9],
a skew-morphism of a finite group A is a bijective mapping
f:A→A fixing the identity element of A and having the property that f(xy)=f(x)fπ(x)(y) for all
x,y∈A, where the integer π(x) depends only on x.
We also refer to the mapping π:A→ℤ as a power function corresponding to f. The original motivation to investigate skew-morphisms of groups come from the characterisation of regular Cayley maps proved by Jajcay and Širáň [9]:
“A Cayley map CM(G;ρ), where G is a group and ρ is a cyclic permutation of the generators is regular if and only if ρ extends to a skew-morphism of G”. Note that the orbit of ρ is by definition closed under taking inverses and the existence of at least
one generating orbit closed under taking inverses is a characteristic feature of skew-morphisms related to Cayley regular maps. Recall
that to define a 2-cell embedding of a connected graph into an orientable surface combinatorially one needs to determine a cyclic permutation of arcs based at v for each vertex v. A Cayley map is an embedding of a Cayley graph into an orientable surface such that a chosen global orientation induces at each vertex
the same cyclic permutation of generators. Cayley maps are vertex-transitive, since each Cayley automorphism of the graph extends to an automorphism of the map. If the group of map automorphisms is regular on arcs, the map itself is called regular. Another geometric motivation for investigation of skew-morphisms can be found in papers [8, 13], where skew-morphisms are studied in connection with highly symmetrical hypermaps and with regular embeddings of complete bipartite graphs.
Although the concept of a skew-morphism
was introduced and investigated in the context of regular Cayley maps and hypermaps, see [3, 5, 8, 9],
Conder, Jajcay and Tucker [4] pointed out that it appeared
already in the context of factorisation of groups. Namely, let G be a finite group having a factorisation G=AB into subgroups A and B with B cyclic and A∩B=1, and let b be a generator of B. Then there exists a bijective mapping
f:A→A well-defined by the equality baB=f(a)B, where a∈A.
It is not hard to show that f is a skew-morphism of A. Moreover,
all skew-morphisms of A arise in this way. Every automorphism of A is a skew-morphism with the power function π(x)=1 for all x∈A; the converse, however, does not hold in general.
In particular, the normality of A in G=AB, is a necessary and sufficient condition for a skew-morphism defined by this factorisation to be an automorphism. There are plenty of factorisations AB, where A is not normal and B is cyclic, and there are many skew-morphisms that are not automorphisms, even in the case when A is cyclic.
Particular examples of skew-morphisms which are not
group automorphisms were given in [1, 4, 12] for abelian groups
and in [4, 14] for dihedral groups. The problem of determining
all skew-morphisms for a given group arises. Since every automorphism
is a skew-morphism, this problem is at least as hard as the problem of
determining Aut(G) for a given group G, which is not an easy task in general. One of the problems one needs to overcome consists in the fact that the composition of two skew-morphisms of G may not be a skew-morphism. Even to determine the skew-morphisms of cyclic groups seems to be a difficult problem.
Problem.
Determine the skew-morphisms of cyclic groups.
In this paper, we make a step towards its solution
by determining all skew-morphisms for cyclic p-groups, where p>2 is an odd prime. The next natural step to accomplish is a determination of skew-morphisms of cyclic 2-groups. A partial solution of the above problem appears in [5], where skew-morphisms containing a generating orbit closed under taking inverses are classified.
In [12], we investigated skew-morphisms of cyclic groups in connection with Schur rings of cyclic groups. Among others we proved the following decomposition theorem.
In what follows we denote by Skew(ℤn) the set of
skew-morphisms of the cyclic additive group ℤn, and by
ϕ Euler’s totient function.
Theorem 1.1 (cf. [12, Theorem 1.1]).
Let n=n1n2, where gcd(n1,n2)=1, and gcd(n1,ϕ(n2))=gcd(ϕ(n1),n2)=1. Then σ∈Skew(Zn) if and only if σ is of the form σ=σ1×σ2, where σ1∈Skew(Zn1) and σ2∈Skew(Zn2).
It was shown in [12, Corollary 4.10] (and reproved in [4, Theorem 6.4]) that the cyclic group ℤp2 has exactly (p-1)(p2-2p+2) skew-morphisms, where p is an odd prime.
In this paper, we generalise this result to the cyclic group ℤpe of order pe with arbitrary e≥2 by proving
Theorem 1.2.
If e≥2 and p is an odd prime, then the cyclic group Zpe has
exactly (p-1)(p2e-1-p2e-2+2)/(p+1) skew-morphisms.
Detailed description of the set Skew(ℤpe) of skew-morphisms can be found in Theorems 5.4 and 5.5,
which are the main results of the paper.
The paper is organised as follows. In the next section we recall some known facts for further use.
Among others we state in Lemma 2.1 that the order of a skew-morphism of a cyclic group ℤn divides nϕ(n). In particular, if n=pe for a prime p, the skew-morphisms split into two classes: the first class contains those skew-morphisms whose order is a power of p, the others form the complement of the first class. In Section 3 we define a two-parametrised family of skew-morphisms si,j and investigate its properties. Although the skew-morphisms of a group do not form a group in general, in the investigated case of cyclic p-groups, the skew-morphism si,j all “live” in a split metacyclic group G=CY of order p2e-1, where C is a normal cyclic subgroup of order pe and Y is cyclic of order pe-1, see Remark 3.2.
In Section 4 the above family is generalised to a family of skew-morphisms si,j,k,l given by four integer parameters i,j,k,l. Finally, in the last section we prove
that every skew-morphism in Skew(ℤpe) is one of
si,j,k,l,
and that the skew-morphisms of the first class are exactly the skew-morphisms si,j=si,j,0,0, see Theorems 5.4 and 5.5. For the purpose of enumeration we clarify in Proposition 4.4 which 4-tuples of parameters determine the same skew-morphism.
A key role in our investigation is played by Huppert’s theorem [7, III.11.5] establishing the structure of the skew product group in case the order of the skew-morphism is a power of p, and also by a result
of King [11] on presentations of metacyclic p-groups.
In the proof that the parametrised family of skew-morphisms
si,j,k,l equals Skew(ℤpe), we use some essential information on automorphisms of split metacyclic
p-groups obtained by Bidwell and Curran [2].
2 Preliminary results
Throughout the paper ℤpe represents the cyclic group of order pe, where p is an odd prime.
Let Sym(ℤpe),Aut(ℤpe) and Skew(ℤpe) denote the set (group) of all permutations, automorphisms, and skew-morphisms of ℤpe, respectively. In this paper we multiply
permutations from the right to the left, that is, if f and g are
in Sym(ℤpe) and x∈ℤpe, then (fg)(x)=f(g(x)).
We set t to denote the translation of ℤpe defined by
t(x)=x+1.
Let s∈Skew(ℤpn) be a skew-morphism with power function π. The skew product group of ℤpe induced by s is the group 〈t,s〉. Notice that the skew product group
factorises as 〈t,s〉=〈t〉〈s〉,
and for every i∈ℤpe,
(2.1)sti=ts(i)sπ(i).
Also, |〈t,s〉|=pe⋅|s|, where |s| is the order of the permutation s. The following
converse also holds (see [12, Proposition 2.2]): if
f∈Sym(ℤpe), then
(2.2)f(0)=0 and |〈t,f〉|=pe⋅|f|⟹f∈Skew(ℤpe).
The next result is in [12, Corollary 3.4], see [4, Theorem 6.1], as well.
Lemma 2.1.
Let s∈Skew(Zn). Then the order |s| of s divides
nϕ(n). Moreover, if gcd(|s|,n)=1 or gcd(ϕ(n),n)=1, then
s is an automorphism of Zn.
In the next lemma we collect a number of properties of skew-morphisms of ℤpe proved in [12].
In fact, part (i) is [12, Lemma 4.4 (iii)], (ii) is
[12, Lemma 4.4 (ii)], and (iii) is [12, Theorem 4.1].
Lemma 2.2.
Let s∈Skew(Zpe) be a skew-morphism of order pi with power function π.
The power sp is also in Skew(ℤpe).
For all x∈ℤpe,
π(x)≡1(modp). In particular,
if s has order p, then s is in Aut(ℤpe).
There exists an automorphism α∈Aut(ℤpe) of
order pi such that the 〈α〉-orbits
coincide with the 〈s〉-orbits.
It is well known that
Aut(ℤpe)≅ℤ(p-1)pe-1,
and the automorphisms of order pi (1≤i≤e-1) are given as
(2.3)x↦(kpe-i+1)x,
where k∈{1,…,pi-1} and p∤k (see, e.g., [10]).
This implies that, for any k∈{0,…,pn-1},
(2.4)gcd((p+1)k-1,pe)=p⋅gcd(k,pe-1).
Lemma 2.3.
Let s∈Skew(Zpe) be a skew-morphism of order pi.
Then
gcd(s(1)-1,pe)=pe-i.
Proof.
The statement is clear if i=0, let i≥1.
According to Lemma 2.2 (iii),
the orbit of 1 under 〈s〉 is equal to the orbit
of 1 under 〈α〉 for some automorphism
α of order pi. By (2.3), this orbit is
Ω:={xpe-i+1:x∈{0,…,pi-1}}.
This in turn implies that 〈s〉 is regular on Ω,
and thus s(1) is not in the orbit of 1 under 〈sp〉.
Since sp is a skew-morphism, see Lemma 2.2 (i), it follows
that s(1)∉{xpe-i+1+1:x∈{0,…,pi+1-1}}.
The lemma is proved.
∎
We end the section with a property of metacyclic p-groups.
Lemma 2.4.
Let G be a metacyclic p-group given by the presentation
G=〈x,y∣xpm=ypn=1,xy=x1+pm-n〉,
where m>n≥1. Then the order of the element xiyj
is equal to max{|xi|,|yj|}, where |xi| and |yj| denote
the order of xi and yj, respectively.
Proof.
We prove the lemma by induction on m. If m=2, then
G is the unique non-abelian group of order p3 and of
exponent p2. In this case the lemma follows by a straightforward
computation.
Let m≥3 and N=〈xpm-1,ypn-1〉.
We claim that N≅ℤp2, N is normal in G. Indeed, setting
z=ypn-1 and w=xpm-1, we have z-1xz=x1wk, for some k. It follows that x-1z-1x=wkz-1∈N, and so conjugation by both x and y fixes N=〈z,w〉.
The factor group G/N admits the presentation
G/N=〈x¯,y¯∣x¯pm-1=y¯pn-1=1,x¯y¯=x¯1+pm-n〉,
where for g∈G, g¯ denotes the image of g
under the natural homomorphism G→G/N.
The lemma holds trivially if either xi=1 or yj=1, hence
we assume below that xi≠1 and yj≠1.
Then we find, using the induction hypothesis, that the order of x¯iy¯j is equal to max{|x¯i|,|y¯j|}=1pmax{|xi|,|yj|}. Therefore, we are done if we show that
x¯iy¯j has order 1p|xiyj|,
where |xiyj| denotes the order of xiyj.
Obviously, xiyj≠1. Let z be an element in 〈xiyj〉 of order p. If z∉N, then 〈N,z〉 is isomorphic to a group of order p3 and of exponent p. This
implies that 〈N,z〉 is not metacyclic, contradicting the
fact that every subgroup of G is metacyclic (cf. [7, III.11.1]).
Therefore, z∈N, and thus x¯iy¯j has order 1p|xiyj|, as it is required.
∎
3 The skew-morphisms si,j
For the rest of the paper let a∈Aut(ℤpe) be the
automorphism defined by
a(x)=(p+1)x.
By (2.3), a has order pe-1.
Consider the permutation taj∈Sym(ℤpe) for some
j∈{0,1,…,pe-1-1}. Recall that
t is the translation t(x)=x+1. By Lemma 2.4, taj
has order pe, and thus it is a full cycle. Therefore, there
exists a unique permutation bj∈Sym(ℤpe) such that bj(0)=0 and the conjugate
(3.1)tbj:=bjtbj-1=taj.
In fact, for x∈ℤpe with x≠0, the permutation bj can be expressed as
(3.2)bj(x)=1+(p+1)j+⋯+(p+1)(x-1)j.
Define the permutations
(3.3)si,j=bj-1aibj,i,j∈{0,1,…,pe-1-1}.
Proposition 3.1.
Every permutation si,j defined in (3.3) is a skew-morphism of Zpe.
Proof.
By definition, the order |si,j|=|ai|.
Assume at first that p∤i, or
equivalently, si,j has order pe-1.
It is clear that si,j(0)=0. Thus by (2.2), it is
enough to show that |〈t,si,j〉|=p2e-1.
We have
|〈t,si,j〉|=|〈t,si,j〉bj|=|〈tbj,si,jbj〉|=|〈taj,ai〉|=p2e-1,
therefore si,j is a skew-morphism. The last equality comes from the following observation: assuming 〈taj〉∩〈ai〉≠1 we get that some power of a centralises t, a contradiction with the definition of a and t.
Now, suppose that i=pki′,p∤i′. As si′,j is a skew-morphism of ℤpe, and
si,j=si′,jpk,
it follows from
Lemma 2.2 (i) that si,j is a skew-morphism too.
∎
Remark 3.2.
The reader might observe that in case a skew-morphism
of the cyclic group of order pe has order pe-1, then the
skew product G has the following canonical presentation:
G=〈x,y∣xpe=ype-1=1,xy=x1+p〉.
In the above notation t=x and conjugation by y is the automorphism a. Besides the canonical factorisation G=〈x〉〈y〉, we have a factorisation G=〈xyj〉〈y〉, for any j. By Lemma 2.4, 〈xyj〉 has order pe, so y gives another skew-morphism of the cyclic group of order pe (generated by different element of order pe). Given two factorisations G=BY=CY, where Y=〈y〉, |B|=|C|=pe for any c∈C there is a unique b∈B such that c=byi. Thus there is a bijection σ:C→B that provides a “change of coordinates”. Let ϕ and ψ be the two skew-morphisms determined by y with respect to the two factorisations CY and BY.
Then ycY=ϕ(c)Y and ycY=yσ(c)Y=ψ(σ(c))Y. The last term is by the factorisation BY. To convert back to CY, we use σ-1, thus ϕ=σ-1ψσ. The skew-morphisms si,j all live in the split metacyclic p-group G and are obtained by left multiplication on 〈xjy〉, to be precise, they are conjugate to yi by the permutation bj.
Notice that the skew-morphism si,j is not uniquely determined by
the parameters i and j. For instance, s0,j is the identity mapping for every j. The rest of the section is devoted to the proof of following theorem.
Theorem 3.3.
Let e≥2, and i,i′,j,j′∈{0,1,…,pe-1-1}. Then
si,j=si′,j′⇔i=i′ and j≡j′(modpe-2/gcd(i,pe-2)).
The theorem will be derived in a sequence of lemmas.
Lemma 3.4.
Let e≥2, and j,j′∈{0,1,…,pe-1-1} such that
j≡j′(modpu) for some u∈{0,1,…,e-1}. Then
for all x,x′∈Zpe with x≡x′(modpe-1-u),
bj′(x)-bj(x′)=bj′(x)-bj(x).
Proof.
We divide the proof into four steps.
Claim 1.
We have bj(pe-1-u)=bj′(pe-1-u).
We start by the observation that if x,y,z∈ℤpe, then
(3.4)(txay)z=txby(z)ayz.
It follows that
(taj)pe-1-u=tbj(pe-1-u)ajpe-1-u,(taj′)pe-1-u=tbj′(pe-1-u)aj′pe-1-u.
Since ajpe-1-u=aj′pe-1-u, Claim 1 is equivalent to
(taj)pe-1-u=(taj′)pe-1-u.
Recall that an arbitrary p-group G is regular if
for all x,y∈G,
(xy)p=xpypc1⋯cr,
where all ci belong to the commutator 〈x,y〉′. In the case when p>2, a sufficient condition for G to be regular is that its commutator subgroup G′ is cyclic (see [7, III.10.2)]). In particular, it follows that 〈t,a〉 is a regular p-group.
Thus by [7, III.10.6],
(taj)pe-1-u=(taj′)pe-1-u
is equivalent to
((taj)-1taj′)pe-1-u=1.
Putting j′-j=j0pu, for some j0, we get
((taj)-1taj′)pe-1-u=(aj0pu)pe-1-u=1.
Claim 2.
We have bj(xpe-1-u)=bj′(xpe-1-u) for all
x∈{0,1,…,pu+1-1}.
Let x∈{0,1,…,pu+1-1}.
Recall that tbj=taj. By this and (3.4),
(txpe-1-u)bj=(taj)xpe-1-u=tbj(xpe-1-u)axpe-1-uj,
(txpe-1-u)bj′=(taj′)xpe-1-u=tbj′(xpe-1-u)axpe-1-uj′.
Thus Claim 2 is equivalent to (txpe-1-u)bj=(txpe-1-u)bj′. Since this holds for x=1 by Claim 1, it also holds
for all x>1.
Claim 3.
We have pe-1-u∣bj(xpe-1-u) for all
x∈{0,1,…,pu+1-1}.
The order of taj is pe.
This implies that (taj)xpe-u-1=tbj(xpe-1)ajxpe-u-1 has order at most pu+1.
As ajxpe-u-1 has order at most pu, it follows from Lemma 2.4 that
tbj(xpe-u-1) has order at most pu+1. This yields Claim 3.
Claim 4.
We have bj′(x′)-bj(x′)=bj′(x)-bj(x) for all
x,x′∈Zpn with x≡x′(modpe-1-u).
Put x′=x+x0pe-1-u and j′=j+j0pu. By (3.2) and Claim 2,
bj′(x′)-bj′(x)=bj′(x)+(p+1)j′xbj′(x0pe-1-u)-bj′(x)
=(p+1)jx(p+1)j0puxbj′(x0pe-1-u)
=(p+1)jx(p+1)j0puxbj(x0pe-1-u).
By (2.3), (p+1)j0pux≡1(modpu+1). Thus we find, using also Claim 3, that
(p+1)j0puxbj(x0pe-1-u)=bj(x0pe-1-u).
Therefore,
bj′(x′)-bj′(x)=(p+1)jxbj(x0pe-1-u)=bj(x′)-bj(x).
This completes the proof.
∎
Lemma 3.5.
Let e≥2 and i∈{0,…,pe-1-1}. Then for all
x∈Zpe,
si,j(x)≡x(modp⋅gcd(i,pe-1)).
Proof.
By (3.3), the order of si,j is pe-1/gcd(i,pe-1).
By Lemma 2.2 (iii), there exists an automorphism α∈Aut(ℤpe) of the same order such that the 〈α〉-orbits coincide with the
〈si,j〉-orbits. It follows that
α∈〈agcd(i,pe-1)〉.
Let x∈ℤpe. Then there exists
some l∈{0,…,pe-1-1} with gcd(i,pe-1)∣l
such that
si,j(x)=al(x),
and hence
(3.5)si,j(x)≡(p+1)lx(modpe).
By (2.4), (p+1)l≡1(modp⋅gcd(l,pe-1)).
Since gcd(i,pe-1)∣gcd(l,pe-1), it follows that
(p+1)l≡1(modp⋅gcd(i,pe-1)). Substituting this
in (3.5), the lemma follows.
∎
Lemma 3.6.
Let e≥2, and i,j,j′∈{0,…,pe-1-1}. Then
(3.6)si,j=si,j′⇔j≡j′(modpe-2/gcd(i,pe-2)).
Proof.
The statement holds if i=0, and hence below we assume that
i>0.
(⇒)
Let si,j=si,j′, and π be the power function of si,j.
By (2.1),
si,jt=tsi,j(1)si,jπ(1)
holds in
〈t,si,j〉. This implies in 〈t,si,j〉bj=〈taj,ai〉,
t(p+1)iai+j=ai(taj)=(taj)si,j(1)(ai)π(1)=tbj(si,j(1))asi,j(1)j+π(1)i.
As bj(si,j(1))=(bjsi,j)(1)=(aibj)(1)=(p+1)i,
we get
(3.7)ai+j=asi,j(1)j+π(1)i.
Since si,j=si,j′, the same argument yields
ai+j′=asi,j′(1)j′+π(1)i. Thus
a(j-j′)(si,j(1)-1)=1,
and hence
(3.8)j≡j′(modpe-1/gcd(si,j(1)-1,pe-1)).
By (3.3), the order of si,j is
pe-1/gcd(i,pe-1). This and Lemma 2.3 yield
gcd(si,j(1)-1,pe)=p⋅gcd(i,pe-1).
Since i≠0, gcd(i,pe-1)=gcd(i,pe-2) and
gcd(si,j(1)-1,pe)=gcd(si,j(1)-1,pe-1), and thus
gcd(si,j(1)-1,pe-1)=p⋅gcd(i,pe-2).
This and (3.8) yield the right side of (3.6).
(⇐)
Put pu=pe-2/gcd(i,pe-2). Then order of ai is
pe-1/gcd(i,pe-2)=pu+1, and hence by (2.3),
(3.9)(p+1)i=zpe-u-1+1 for some z∈{1,…,pu+1-1},p∤z.
For two arbitrary mappings f,g:ℤpe→ℤpe, their sumf+g, differencef-g, and productfg are the mappings from ℤpe to ℤpe defined in
the usual way, that is, for x∈ℤpe,
(f+g)(x)=f(x)+g(x),(f-g)(x)=f(x)-g(x),(fg)(x)=f(g(x)).
Let j′=j+j0pu, for some j0. For every x∈ℤpe with
x≠0,
((p+1)j0pux-1)∣((p+1)j′x-(p+1)jx).
Therefore, pu+1∣((p+1)j′x-(p+1)jx).
This and (3.2) imply
(bj′-bj)(x)≡0(modpu+1) for all x∈ℤpe.
This and (3.9) yield
ai(bj′-bj)=bj′-bj.
Also, by Lemma 3.5,
si,j(x)≡x(modpe-u-1) for all x∈ℤpe.
Therefore, by Lemma 3.4,
(bj′-bj)(si,j(x))=(bj′-bj)(x)for allx∈ℤpe.
Consequently, (bj′-bj)si,j=bj′-bj=ai(bj′-bj). Thus
aibj′-aibj=bj′si,j-bjsi,j=bj′si,j-aibj;
and we get si,j=bj′-1aibj′=si,j′.
∎
Theorem 3.3 follows from Lemma 3.6 and
the following lemma.
Lemma 3.7.
If si,j=si′,j′, then i=i′.
Proof.
Let si,j=si′,j′, and let π be the power function of si,j.
We prove the lemma by induction on the order of si,j.
The statement holds obviously if si,j is the identity permutation,
that is, if i=0. Thus for the rest of the proof assume that i≠0.
By (3.7),
a(si,j(1)-1)j=ai(1-π(1)) and a(si,j(1)-1)j′=ai′(1-π(1)).
Then
sip,j=si,jp=si′,j′p=si′p,j′.
Thus by the
induction hypothesis, ip≡i′p(modpe-1).
As p∣(π(1)-1), see Lemma 2.2 (ii), we conclude
i(1-π(1))=i′(1-π′(1))(modpe-1).
Thus the above equalities reduce to
j≡j′(modpe-1/gcd(pe-1,si,j(1)-1)).
Since it has already been shown that
gcd(si,j(1)-1,pe-1)=p⋅gcd(pe-2,i), it follows that
j≡j′(modpe-2/gcd(pe-2,i)).
Thus by Lemma 3.6, si′,j′=si′,j, and we get
si,j=si′,j′=si′,j. It is obvious that this implies that
i=i′.
∎
4 The skew-morphisms si,j,k,l
For the rest of the paper we set b to be an automorphism of
ℤpe of order p-1.
Define the permutations
(4.1)si,j,k,l=bj-1aibkblbj,
where the integers i,j,k,l satisfy the following
conditions:
i,l∈{0,…,pe-1-1}, k∈{0,…,p-2}, j∈{0,…,pe-2-c-1}, where
pc=gcd(i,pe-2).
If i≠0 and k≠0, then pc∣j and
pmax{c,e-2-c}∣l.
A 4-tuple (i,j,k,l) of integers satisfying conditions (C0), (C1) and (C2) will be called admissible.
Note that the permutations si,j,k,l include all
skew-morphisms in the form si,j. Namely, given any
skew-morphism si,j, in view of Theorem 3.3 we may assume that j<pe-2-c for c=gcd(i,pe-2-), and thus by
(3.3) and (4.1), si,j=si,j,0,0.
Before we prove that all permutations si,j,k,l are
skew-morphisms, we give two lemmas.
Lemma 4.1.
Let G be a split metacyclic p-group given by the presentation
G=〈x,y∣xpm=ypn=1,xy=x1+pm-n〉,
where m>n≥1. Then the automorphisms in Aut(G) are
the mappings θu,v,w, where
u,v∈{0,…,pm-1} with p∤u, pm-n∣v, and
w∈{0,…,pn-1} with p2n-m∣w whenever
2n>m, defined as
θu,v,w(xiyj)=xui+vjywi+j for all xiyj∈G.
In particular, |Aut(G)|=(p-1)pm-1+n+min{n,m-n}.
Proof.
This is a corollary of the more general result
[2, Theorem 3.1].
∎
Lemma 4.2.
With the notation of Lemma 4.1, let u∈{0,…,pm-1},
p∤u such that u≠1, and as a unit of the ring Zpm, u has order d with p∤d.
Then the automorphism θu,0,w has order d.
Proof.
A straightforward computation gives
(θu,0,w)k(xiyj)=xukiywi(1+u+⋯+uk-1)+j.
Therefore, the order |θu,0,w|≥d.
Also, pm∣(ud-1). On the other hand, since the order of
the unit u≠1 is not divisible by p, it follows that p∤(u-1) as well.
We conclude from these that pm∣(1+u+⋯+ud-1). Using also that n<m, we find
(θu,0,w)d(xiyj)=xiyj, and thus
|θu,0,w|=d.
∎
Theorem 4.3.
Every permutation si,j,k,l defined in (4.1) is a
skew-morphism of Zpe.
Proof.
Let si,j,k,l be a permutation defined in (4.1).
If k=0, then si,j,k,l=si,j and the statement holds.
For the rest of the proof it is assumed that k≠0.
Let i=0. Then j=0 by (C0) and l=0 by (C1). We get
si,j,k,l=bk, which is an automorphism in Skew(ℤpe).
Now, suppose that i≠0.
Since k≠0, we have tbk=tu for some
u∈{2,…,pe-1}, p∤u, and as unit of ℤpe,
u has order d with p∤d.
Let pc=gcd(i,pe-2).
Let G=〈taj,ai,bkbl〉 and
P=〈taj,ai〉. It follows from (C2) that
P=〈t,ai〉 and there exists a1∈〈ai〉
such that P admits the presentation
P=〈t,a1∣tpe=(a1)pe-1-c=1,ta1=t1+pc+1〉.
Clearly, a1=ai′ for a some i′ satisfying
gcd(i′,pe-2)=pc. This and (C2) yield pe-2/gcd(i′,pe-2)∣l, and so si′,l=si′,0=ai′ follows from Theorem 3.3. In other words, bl commutes with a1=ai′.
Now, we can write
(4.2)tbkbl=tual and (a1)bkbl=a1.
The group 〈a1〉 is equal to 〈apc〉.
On the other hand, by (C2), pc∣l, and so
al=a1w for some w∈{0,…,pe-1-c-1}.
This and (4.2) yield that P is normal in G.
We claim that bkbl acts on P by conjugation as the
automorphism θu,0,w described in Lemma 4.1.
In fact, we have to show that pe-2-2c divides w whenever
2(e-1-c)>e. In order to see that this indeed holds, observe that
al=a1w=ai′w, gcd(i′,pe-2)=pc,
and finally pe-2-c∣l, by (C2).
Since bkbl fixes 0 and P is transitive on ℤpe,
Z〈bkbl〉(P)=1. Equivalently, bkbl acts
faithfully on P, in particular, |bkbl|=|θu,0,l|.
Now, by Lemma 4.2, |bkbl|=d. Thus
|G|=|P|⋅d. This implies that the stabiliser G0 of 0 in G
has order |G0|=pe-1-cd.
Also, as ai commutes with bkbl, we find
|aibkbl|=pe-1-cd=|G0|,
implying
that G0=〈aibkbl〉, and G factorises as
G=〈taj〉〈aibkbl〉.
Therefore, the conjugate group G(bj)-1 factorises as
G(bj)-1=〈t〉〈si,j,k,l〉,
and hence si,j,k,l is a skew-morphism by (2.2).
∎
Proposition 4.4.
Every permutation si,j,k,l in (4.1) is uniquely determined
by the admissible 4-tuple (i,j,k,l). In particular, there is a bijection between the set of admissible 4-tuples of integers and the set of skew-morphisms
si,j,k,l.
Proof.
Suppose that si,j,k,l and si′,j′,k′,l′ are two skew-morphisms
defined in (4.1) for which si,j,k,l=si′,j′,k′,l′.
We are going to prove that
(i,j,k,l)=(i′,j′,k′,l′).
It follows that these skew-morphisms have order
pe-1(p-1)gcd(i,pe-1)gcd(k,p-1)=pe-1(p-1)gcd(i′,pe-1)gcd(k′,p-1).
In particular, gcd(i,pe-1)=gcd(i′,pe-1).
Let i=0 or i′=0. Then we get i=i′=0, hence j=j′=0,
and l=l′=0, by condition (C1). Thus bk=s0,0,k,0=s0,0,k′,0=bk′, implying that k=k′, and so
(i,j,k,l)=(i′,j′,k′,l′).
Now, suppose that none of i and i′ is equal to 0.
Let gcd(i,pe-2)=pc (so gcd(i′,pe-2)=pc as well).
By (3.3) and (4.1),
si,jbj-1bkblbj=si′,j′bj′-1bk′bl′bj′.
Both si,j and si′,j′ have order pe-1-c≠1,
and both bj-1bkblbj and bj′-1bk′bl′bj′
have the same order not divisible by p. We have proved above
that
[si,j,bj-1bkblbj]=[si′,j′,bj′-1bk′bl′bj′]=1
also hold, and we deduce from these that
si,j=si′,j′ and
bj-1bkblbj=bj′-1bk′bl′bj′.
Since both j and j′ are in {0,…,pe-2-c-1}, it follows by Theorem 3.3 that i=i′ and j=j′. This implies that
bkbl=bk′bl′. From this by (3.2), we obtain bk(1)=(bkbl)(1)=(bk′bl′)(1)=bk′(1). This in turn implies that k=k′ and
bl=bl′. Then, by (3.2) again,
(p+1)l=bl(2)-1=bl′(2)-1=(p+1)l′, from which l=l′.
This completes the proof of the proposition.
∎
and
Roman Nedela
