Nonuniqueness of semidirect decompositions for semidirect products with directly decomposable factors and applications for dihedral groups

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by  Peteris Daugulis

NONUNIQUENESS OF SEMIDIRECT DECOMPOSITIONS FOR
6
SEMIDIRECT PRODUCTS WITH DIRECTLY DECOMPOSABLE
1
0 FACTORS AND APPLICATIONS FOR DIHEDRAL GROUPS
2

p PETERIS DAUGULIS∗
e
S
Abstract. Nonuniqueness of semidirect decompositions of groups is an insufficiently studied
8
questionincontrasttodirectdecompositions. Weobtainsomeresultsabout semidirectdecomposi-

tionsforsemidirectproductswithfactorswhicharenontrivialdirectproducts. Wedealwithaspecial
]
R case of semidirect product when the twisting homomorphism acts diagonally on a direct product,
G as well as forthe case whenthe extending groupis adirectproduct. We giveapplications of these
resultsinthecaseofgeneralizeddihedralgroupsandclassicdihedralgroupsD2n. ForD2n wegive
.
h acompletedescriptionofsemidirectdecompositions andvaluesofminimalpermutationdegrees.
t
a
m Key words. semidirectproduct, directproduct,diagonalaction, generalizeddihedralgroup
[

3 AMS subject classifications. 20E22,20D40.
v
6
6 1. Introduction.
3
5
1.1. Background. Theaimofthisarticleistostudysemidirectdecompositions
0
. of groups both in general and special cases.
1
0
By the well known Krull-Remak-Schmidt theorem the multiset of isomorphism
6
1 types of indecomposable direct factors for groups satisfying ascending and descend-
: ing chain conditions on normal subgroups does not depend on the order of factors.
v
i Thus direct decompositions of such groups, e.g. finite groups, may be considered
X
understood.
r
a
FewresultsofthistypeareknownforsemidirectandZappa-Szepdecompositions.
One can mention the Schur-Zassenhaus theorem as an example.
We consider cases when the base group or the extending group is a direct prod-
uct. We present a general result which allows to characterize some semidirect de-
compositions in the case when the base group is a direct product and the twisting
homomorphism acts diagonally, Prop.2.4. Finally we obtain a nonuniqueness result
ofsemidirectdecompositioninthecasewhentheextendinggroupisadirectproduct,
Prop.2.6. We give applications of some of these results in the case which is relatively
easy to understand - finite dihedral groups, both classic and generalized.
∗Department of Mathematics, Daugavpils University, Daugavpils, LV-5400, Latvia (pe-
[email protected]).
1
2 P.Daugulis
We use traditional multiplicative notation for general groups and additive nota-
tion for abeliangroups. In this article the dihedralgroupof order m=2n is denoted
by D - D = ha,x|an = e,x2 = e,xax = a−1i, m = 2n. For any m|n we usually
m m
identify Z with the corresponding subgroup of Z . Q denotes the dicyclic group
m n m
of order m=4k - Q =ha,x|a2k =e,x2 =ak,x−1ax=a−1i.
m
The cyclic group of order m is denoted by Z , in additive notation we assume
m
that Z =h1i. In this article we identify elements of Z and correspondingminimal
m m
nonnegative integers. We use this identification for powers of group elements. For
example, if r ∈Z and r ≡2(mod 3), then ar =a2 for any group element a.
3
1.2. Basic facts about semidirect products.
We remind the reader that an external semidirect product of groups N (base
group) and H (extending group) is the group N ⋊ H =(N ×H,·) where the group
ϕ
product is defined on the Cartesian product N ×H using a group homomorphism
(twisting homomorphism) ϕ ∈ Hom(H,Aut(N)) as follows: (n ,h ) · (n ,h ) =
1 1 2 2
(n ϕ(h )(n ),h h ). Sets N = N × {e } and H = {e } × H are subgroups in
1 1 2 1 2 H N
N ×H.
e e
A group G is an internal semidirect product of its subgroups N and H if N is a
normal subgroup, G = NH and N ∩H = {e}. If a group G is finite then for G to
be an internal semidirect product NH is equivalent to 1) N being normal in G, 2)
|N|·|H|=|G| and 3) N∩H ={e}. In the internalcase the twisting homomorphism
H →Aut(N) is given by the map h7→(n→hnh−1), for any n∈N, h∈H.
Both expressions will be called semidirect decompositions of G. If the twisting
homomorphismisnotdiscussed,weomititandusethenotation⋊. Weconsiderdirect
product to be a special case of semidirect product with the twisting homomorphism
being trivial. For relevant treatment see [5], [6].
A nontrivial semidirect product may admit more than one semidirect decompo-
sition. Examples are abundant starting from groups of order 8.
Example 1.1. Twisting homomorphisms are not given in these examples.
D ≃Z ⋊Z ≃Z2⋊Z . Σ ≃A ⋊Z ≃Z2⋊Σ .
8 4 2 2 2 4 4 2 2 3
There are semidirect products such that Z ⋊Q ≃Q , but Q ⋊Z ≃SL(2,F ).
3 8 24 8 3 3
On the other hand, there is a group G of order 32, such that G ≃ D ⋊Z2 ≃
32 32 8 2
Z2⋊D .
2 8
Finally, there is a group G of order 24 which can be decomposed in 5 different
24
ways: G ≃Z ⋊D ≃Z2⋊Z ≃D ⋊Z ≃(Z ×Z2)⋊Z ≃Q ⋊Z .
24 3 8 2 3 12 2 3 2 2 12 2
Nonuniqueness ofsemidirectdecompositions andapplications fordihedralgroups 3
2. Main results.
2.1. Diagonal semidirect products.
2.1.1. Automorphisms of direct products. We introduce a linear algebra
style notation for direct products of groups.
g
LetG=G ×G . Encodetheelement(g ,g )asacolumn 1 . Ifϕ∈Aut(G),
1 2 1 2
" g2 #
g ϕ (g ,g )
then ϕ( 1 ) = 1 1 2 . One can theck, that for all relevant parameter
" g2 # " ϕ2(g1,g2) #
values ϕ satisfy the following properties:
i
1. ϕ (ab,e)=ϕ (a,e)ϕ (b,e),
i i i
2. ϕ (e,ab)=ϕ (e,a)ϕ (e,b),
i i i
3. ϕ (a,b)=ϕ (a,e)ϕ (e,b)=ϕ (e,b)ϕ (a,e),
i i i i i
Define ϕ (g ) = ϕ (g ,e), ϕ (g ) = ϕ (e,g ), ϕ (g ) = ϕ (g ,e), ϕ (g ) =
11 1 1 1 12 2 1 2 21 1 2 1 22 2
ϕ (e,g ), for all g ∈ G . All functions ϕ are group homomorphisms. Thus
2 2 i i ij
ϕ (g ,g )=ϕ (g ,e)ϕ (e,g )=ϕ (g )ϕ (g ).
i 1 2 i 1 i 2 i1 1 i2 2
g ϕ (g ) ϕ (g )
We can encode action of ϕ as follows: ϕ( 1 )= 11 1 12 2 . Thus
" g2 # " ϕ21(g1) ϕ22(g2) #
an automorphism ϕ∈Aut(G ×G ) is determined by 4 group homomorphisms ϕ :
1 2 ij
G →G .
j i
Definition 2.1. We call ϕ ∈ Aut(G ×G ), G 6= {e}, G 6= {e}, a diagonal
1 2 1 2
automorphism if ϕ and ϕ are trivial homomorphisms.
12 21
Definition 2.2. We call (G ×G )⋊ H a diagonalsemidirect product if ϕ(h)
1 2 ϕ
is a diagonal G × G -automorphism for any h ∈ H. Explicitly, there are group
1 2
homomorphisms ϕ (h):G →G such that ϕ(h)(g ,g )=(ϕ (h)(g ),ϕ (h)(g )).
ii i i 1 2 11 1 22 2
Remark 2.3. Ker(ϕ) = Ker(ϕ )∩Ker(ϕ ). Note that G may not be inde-
11 22 i
composable as direct factors. Described encodings and diagonal semidirect products
can be generalized to cases when the base groups splits into an arbitrary finite num-
ber of direct factors. Similar encodings can be used considering internal semidirect
products.
2.1.2. Semidirect decompositions of diagonal semidirect products.
We presenta propositionshowingnonuniqueness ofsemidirectdecompositionfor
diagonal semidirect products. Vaguely speaking, any direct factor of the base group
which is invariant with respect to the initial twisting homomorphism can be moved
4 P.Daugulis
tothe extendinggroup(nonnormalsemidirectfactor)toenlargeit. The newtwisting
homomorphism is such that the moved direct factor acts trivially on the remaining
part of the base group.
Proposition 2.4. Let N ,N ,H be groups. Let G=(N ×N )⋊ H be a diag-
1 2 1 2 ϕ
onal semidirect product, ϕ(h)(g ,g ) = (ϕ (h)(g ),ϕ (h)(g )). Then the following
1 2 11 1 22 2
statements hold.
1. G≃N ⋊ (N ⋊ H), for some Φ ∈Hom(N ⋊ H,Aut(N )).
1 Φ11 2 ϕ22 11 2 ϕ22 1
^
2. Ker(Φ )=N Ker(ϕ ).
11 2 11
g g e
3. If ϕ (h) = id , for any h ∈ H, i.e. ϕ(h)( 1 ) = 1 ,
11 fN1 " g2 # " e ϕ22(h)(g2) #
then G≃N ×(N ⋊ H).
1 2 ϕ22
Proof.
1. ConsiderN ⋊ (N ⋊ H)whereΦ (n ,h)=ϕ (h). Itisdirectlychecked
1 Φ11 2 ϕ22 11 2 11
that Φ ∈Hom(N ⋊H,Aut(N )). We will prove that
11 2 1
(N ×N )⋊ H ≃N ⋊ (N ⋊ H).
1 2 ϕ 1 Φ11 2 ϕ22
Defineabijectivemapf :(N ×N )⋊ H →N ⋊ (N ⋊ H)byf((n ,n ),h)=
1 2 ϕ 1 Φ11 2 ϕ22 1 2
(n ,(n ,h)), for all n ∈N , h∈H. We prove that f is a group homomorphism.
1 2 i i
Let a,a′ ∈N , b,b′ ∈N , h,h′ ∈H. We have that
1 2
((a,b),h)·((a′,b′),h′)=((a,b)ϕ(h)(a′,b′),hh′)=
=((a,b)(ϕ (h)(a′),ϕ (h)(b′)),hh′)=((aϕ (h)(a′),bϕ (h)(b′)),hh′).
11 22 11 22
On the other hand,
(a,(b,h))·(a′,(b′,h′))=(aΦ (b,h)(a′),(b,h)·(b′,h′))=
11
=(aϕ (h)(a′),(bϕ (h)(b′),hh′)). We see that f is a group isomorphism.
11 22
^
2. Ker(Φ )={(n ,h)|h∈Ker(ϕ )}=N Ker(ϕ ).
11 2 11 2 11
3. In notations given above, ϕ (h) = id implies Φ (n ,h) = id , for any
11 fN1 11 2 N1
n ∈N , h∈H. Thus it is the direct product.
2 2
g g2 e
Example 2.5. Let G=(Z ×Z )⋊ Z , where ϕ(1)( 1 )= 1 . In
7 9 ϕ 3 " g2 # " e g24 #
g 2g 0
additive notation this can be simplified as follows. ϕ(1)( 1 ) = 1 =
" g2 # " 0 4g2 #
Nonuniqueness ofsemidirectdecompositions andapplications fordihedralgroups 5
2 0 g
1
. G can be defined as the subgroup of Σ generated by three permu-
16
0 4 g
" #" 2 #
tations:
a) (1,...,7) (generating Z ),
7
b) (8,...,16) (generating Z ) and
9
c) (1,2,4)(3,6,5)(8,11,14)(9,15,12)(generating action of Z on Z ×Z ).
3 7 9
Z7 Z9
We |have th{azt G≃}Z| ⋊(Z {⋊z Z )≃}Z ⋊(Z ⋊ Z ).
7 9 4 3 9 7 2 3
2.2. Directly decomposable extending groups.
We show that a direct factor of the extending group can be transferred to the
base group.
Proposition 2.6. Let N,H ,H be groups Then
1 2
N ⋊ (H ×H )≃(N ⋊ H )⋊ H ,
ϕ 1 2 ϕ1 1 ϕ2 2
where ϕ (h )(n) = ϕ(h ,e )(n) and ϕ (h )(n,h ) = (ϕ(e ,h )(n),h ), for all
1 1 1 H2 2 2 1 H1 2 1
n∈N, h ∈H .
i i
Proof. It is checked that ϕ are group homomorphisms.
i
We prove that the map f : N ⋊ (H × H ) −→ (N ⋊ H ) ⋊ H given by
1 2 1 2
f(n,(h ,h ))=((n,h ),h ) is a group homomorphism.
1 2 1 2
Let n,n′ ∈N, h ,h′ ∈H .
i i i
Consider the product (n,(h ,h ))·(n′,(h′,h′)) in N ⋊ (H ×H ):
1 2 1 2 ϕ 1 2
(n,(h ,h ))·(n′,(h′,h′))=(nϕ(h ,h )(n′),(h h′,h h′)).
1 2 1 2 1 2 1 1 2 2
Consider the product ((n,h ),h )·((n′,h′),h′) in (N ⋊H )⋊H :
1 2 1 2 1 2
((n,h ),h )·((n′,h′),h′)=(((n,h )ϕ (h )(n′,h′)),h h′)=
1 2 1 2 1 2 2 1 2 2
=(((n,h )(ϕ(e,h )(n′),h′)),h h′)=((nϕ (h )(ϕ(e,h )(n′)),h h′),h h′)=
1 2 1 2 2 1 1 2 1 1 2 2
=((nϕ(h ,h )(n′),h h′),h h′).
1 2 1 1 2 2
We see that both products have equalcorrespondingcomponents and thus f is a
group isomorphism.
Example 2.7. Let G=Z ⋊ (Z ×Z ) where ϕ(x,y)(1)≡(−1)x2y(mod 7). G
7 ϕ 2 3
can be defined as the subgroup of Σ generated by three permutations:
7
a) (1,...,7) (generating Z ),
7
6 P.Daugulis
b) (1,6)(2,5)(3,4) (generating action of Z on Z ) and
2 7
c) (1,2,4)(3,6,5) (generating action of Z on Z ).
3 7
Then G≃D2·7⋊Z3 ≃(Z7⋊Z3)⋊Z2.
2.3. Applications.
2.3.1. Generalized dihedral groups.
We remind the reader that an external semidirect product D(A) = A⋊ Z is
ϕ 2
called generalized dihedral group provided 1) A is abelian and 2) ϕ(1)(g) = −g for
any g ∈A, in additive notation. We can also denote D(A) by A⋊−1Z2.
Using the well known classification of finite abelian groups we can assume that
A = n Z . We use linear algebra style encoding - we encode (g ,...,g ) ∈ A
i=1 mi 1 n
g
L 1
as a column vector ... . Notations introduced in section 2.1.1 are modified for
 
g
n
 
additive groupnotation. The actionofthe twisting homomorphismis givenby scalar
g (−g ) 0 0 g
1 1 1
or matrix multiplication: ϕ(1)( ... ) = 0 ... 0 = − ... =
     
g 0 0 (−g ) g
n n n
     
g      
1
(−E )· ... , where E is the n×n identity matrix.
n   n
g
n
 
 
Remark 2.8. Generalized dihedral groups are diagonal semidirect products with
an injective twisting homomorphism.
Proposition 2.9. Let A= n Z , let A=A ⊕A , where A = n1 Z ,
i=1 mi 1 2 1 i=1 mi
A = n Z . Then
2 i=n1+1 mi L L
L D(A)≃A1⋊(A2⋊−1Z2)=A1⋊D(A2)≃A1⋊D(A/A1).
Proof. D(A) = (A ⊕A )⋊ Z , where ϕ(1)(g) = −g, for any g ∈ A. Thus
1 2 ϕ 2
ϕ(g ,g )=(−g ,−g ), for any g ∈G . It follows that D(A) is a diagonalsemidirect
1 2 1 2 i i
productwithrespecttoA ⊕A decomposition. AccordingtoProposition2.4wehave
1 2
thatD(A)≃A ⋊ (A ⋊ Z )=A ⋊D(A ),whereΦ (g ,1)(g )=ϕ (1)(g )=
1 Φ11 2 ϕ22 2 1 2 11 2 1 11 1
−g .
1
Example 2.10. Let G = D(Z ⊕Z ). G can be defined as a subgroup of Σ
3 5 8
generated by permutations (1,2,3), (4,5,6,7,8) and (1,2)(4,7)(5,6). Then G≃Z ⋊
3
D2·5 ≃Z5⋊D2·3.
Nonuniqueness ofsemidirectdecompositions andapplications fordihedralgroups 7
2.3.2. Dihedral groups.
Classic dihedralgroups are special cases of generalizeddihedral groups when the
base group is a cyclic group. We give a complete description of semidirect decompo-
sitions of D using both Proposition 2.4 and ad hoc computations.
2n
We use a classical presentation of dihedral groups:
D =ha,x|an =e,x2 =e,xax=an−1i=hai∪haix.
2n
We note that D ≃Z and D ≃Z ×Z , in all other cases D is nonabelian.
2 2 4 2 2 2n
Subgroups. Let n ∈ N, n ≥ 3, d ∈ N, d|n, m = n. It is known that D has the
d 2n
following subgroups, see [2].
1. For each m∈N such that m|n there is a subgroup
Am =hamni=hadi={e,ad,a2d,...,a(m−1)d}≃Zm.
A ED for all m. The number of such subgroups is d(n) (the number of
m 2n
natural n-divisors).
2. For each m∈N such that m|n and each r∈Zmn =Zd there is a subgroup
B2m,r =hamn,arxi=had,arxi=hAm,Am(arx)i≃D2m.
Note that r ∈ Zn is identified with an integer as described in the introduc-
m
tion.
The number of such subgroups is σ(n) (the sum of natural n-divisors).
If 2|n then B ED . In all other cases, if 1<m<n then B 6ED .
n,r 2n 2m,r 2n
Classical decompositions. It known that D ≃ Z ⋊ Z where the twisting
2n n ϕ 2
homomorphism is ϕ(1)(g)=−g. In internal terms, D =A ⋊B , for all r ∈Z .
2n n 2,r n
If 2|n and 46|n, then D ≃D ×Z , or, in internal terms, D =B ×A . where
2n n 2 2n n,r 2
r ∈ Z . Again, note, that second indices of B-type subgroups can be interpreted as
2
both integers and residues.
External semidirect decompositions of D . Using Proposition 2.4 we get an ex-
2n
austive description of external semidirect decompositions of D .
2n
Proposition 2.11.
1. D2n ≃ Zm ⋊ϕD2mn, for any m ∈ N, m|n, such that GCD(m,mn) = 1. ϕ is
defined as follows: if D2n =ha,x|amn =e,x2 =e,xax=a−1i then ϕ(a)(1)=
m
1 and ϕ(x)(1)=−1.
8 P.Daugulis
2. D ≃ D ⋊ Z , if n = 2αq, α ∈ N. ϕ is defined as follows: if D =
2n n ϕ 2 n
ha,x|an2 =e,x2 =e,xax=a−1i then ϕ(1)(a)=a−1 and ϕ(1)(x)=ax.
3. If 2|n and 46|n then
D ≃D ×Z .
2n n 2
4. There are no other nontrivial external semidirect decompositions of D in
2n
the following sense. If D ≃ X ⋊Y, |X| > 1, |Y| > 1, then there are two
2n
poosibilities:
a) X =Zm and Y =D2mn, where m|n, GCD(m,mn)=1 or
b) X =D and Y =Z , if 2|n.
n 2
Proof. Statements 1.,2.,3. are provedby exhibiting a suitable internal semidirect
decomposition.
1. We usethe primarydecompositiontheoremforcyclicgroups: ifn= k pαi,
i=1 i
then Zn ≃ ki=1Zpαii. The statement follows from Proposition 2.4. NQote that
Ker(ϕ)=hai.
L
Alternatively, we prove the same statement using the information about D -
2n
subgroups. We show that if GCD(m,mn)=1 then D2n =Am⋊B2mn,r.
We have that Am E D2n and |Am|·|B2mn,r| = 2n = |D2n|. Am ∩B2mn ≤ hamni.
Considering subgroups of hamni it follows that Am∩B2n = {e}. Thus D2n = Am⋊
m
B2mn,r ≃Zm⋊ϕD2mn. Adirectcomputationshowsthatϕisasstated: (am)ad(a−m)=
ad, (arx)ad(arx)=a−d.
Notethatif2|nand46|nthenA ∩B ={e},r ∈Z ,henceD =A ×B ≃
2 n,r 2 2n 2 n,r
Z × D . In this case there are no nontrivial semidirect decompositions of type
2 n
Z ⋊D .
2 n
2. This case is not covered by Proposition 2.4, we show directly that D =
2n
B ⋊B .
n,0 2,1
If 2|n, then B E D . |B |·|B | = |D |. It can be checked that B ∩
n,0 2n n,0 2,1 2n n,0
B ={e}: B =ha2,xi, B =haxi.
2,1 n,0 2,1
Thus D ≃ D ⋊Z . A direct computation shows that ϕ is as stated: (ax)a2
2n n 2
(ax) = a−2 (the generator a2 gets inverted), (ax)x(ax) = a2x (the generator x gets
multiplied by the other generator a2).
3. Using Proposition 2.4 we see that D = D(Z ) = (Z ⊕...)⋊Z ≃ Z ×
2n n 2 2 2
D(Z /Z )≃Z ×D .
n 2 2 n
It can also be proved using the list of subgroups. We remind that D =B ×
2n n,0
A ≃ D ×Z for the following reasons. Both subgroups are normal. |B |·|A | =
2 n 2 n,0 2
Nonuniqueness ofsemidirectdecompositions andapplications fordihedralgroups 9
|D2n|. Bn,0 =ha2,xi, A2 =han2i, n2 is odd, therefore Bn,0∩A2 ={e}.
4. Consider all possible internal semidirect decompositions of D .
2n
If D =X⋊Y then X must be a normalsubgroupof D therefore X must be
2n 2n
A or B with 2|n.
m n,r
If X = Am then Y must be Bm′,r ir order to generate D2n, with m′ = 2mn.
Am∩B2mn,r ={e} iff GCD(n, mn)=1.
Let X = B with 2|n, r ∈ Z . There are n+1 subgroups of D having order
n,r 2 2n
2: B2,r , r ∈ Zn and A2 = han2i. For any n such that 2|n this gives a semidirect
decomposition of type D ⋊Z . If 4 6 |n then A ∩B = {e} which gives a direct
n 2 2 n,r
decomposition D ×Z .
n 2
Remark 2.12. In terms of prime factorization the condition GCD(m, n) = 1
m
is equivalent to the fact that m and n are products of full prime powers of the prime
m
factorization of n. Existence of many members of this family also follows from Schur-
Zassenhaus theorem. If m|n and GCD(m, n) = 1 then GCD(|A |,|D /A |) = 1,
m m 2n m
D2n/Am ≃D2n and, hence D2n ≃Am⋊D2n.
m m
Remark 2.13. Note that there are at most 2 external semidirect decompositions
when n is a prime power:
1. if n=pα, p an odd prime, then there is only one (classical) external semidi-
rect decomposition: D2pα ≃Zpα ⋊Z2,
2. if n = 2α, α ≥ 3, then there are two external semidirect decompositions:
D2·2α ≃Z2α ⋊Z2 ≃D2α ⋊Z2.
Remark 2.14. The image of the twisting homomorphism in each case of a
proper semidirect product is isomorphic to Z . If the extending group is not Z , then
2 2
the twisting homomorphism is not injective.
Example 2.15.
External semidirect decompositions of D2·30:
D ≃Z ⋊Z ≃Z ⋊D ≃Z ⋊D ≃Z ⋊D ≃Z ⋊D ≃Z ⋊D ≃
60 30 2 6 10 10 6 15 4 3 20 5 12
≃D ⋊Z ≃D ×Z .
30 2 30 2
Internal semidirect decompositions of D . We now describe all internal semidi-
2n
rect decompositions of D .
2n
10 P.Daugulis
Proposition 2.16. Let n∈N.
1. If m∈N, m|n, is such that GCD(m, n)=1, then
m
D2n =Am⋊B2mn,r,
for all r ∈Z .
m
2. If n=2αq, α∈N, then
D =B ⋊B =B ⋊B
2n n,0 2,r1 n,1 2,r0
where r ∈Z , r ≡i(mod 2).
i n i
3. If 2|n and 46n then D =B ×A and D =B ×A .
2n n,0 2 2n n,1 2
4. There are no other internal semidirect decompositions of D .
2n
Proof.
1. We look for internal semidirect decompositions of D2n in form Am ⋊Bm′,r.
We must have m′ = 2mn and r ∈ Zm. Am ∩B2mn,r = {e} iff GCD(m,mn) = 1. Thus
D2n =Am⋊B2mn,r for all m such that GCD(m,mn)=1 and all r ∈Zm are the only
possible decompositions of this kind.
2. We look for internal semidirect decompositions of D2n in form Bm,r⋊Bm′,r′.
WemusthaveB ED therefore2|n,m=nandr∈Z ,thuswehavetwopossible
m,r 2n 2
decomposition series: Bn,0⋊B2,r′ and Bn,1⋊B2,r′′. To ensure trivial intersections
of semidirect factors we must have r′ ≡1(mod 2) and r′′ ≡0(mod 2).
3. If 2|n and 4 6 n then B ∩A = B ∩A = {e} where all subgroups are
n,0 2 n,1 2
normal.
4. It follows from the previous arguments.
Permutation representations of dihedral groups. Finally we find minimal degrees
of faithful permutation representations of D . If n is not a prime power then these
2n
numbersaresmallerthandegreesofclassicalpermutationrepresentationsofdihedral
groups. Thisisaconsequenceof2.11andKarpilovskyboundsforfiniteabeliangroups
[4].
Letµ(G)betheminimalfaithfulpermutationrepresentationdegreeofG,i.e. the
minimal n ∈ N such that there is an injective group homomorphism G → Σ . It is
n
known that for finite groups G,H and a group homomorphism ϕ : H → Aut(G) we
havethatµ(G⋊ H)≤|G|+µ(H). If, additionally, ϕisinjective,then µ(G⋊ H)≤
ϕ ϕ
|G|.
Proposition 2.17. Let n = pαi be the prime factorization of n ∈ N. Then
i i
µ(D )= pαi.
2n i i Q
P