Evaluación y requisitos para la obtención de la licencia de funcionamiento 51

We now look at a few examples of groups that have the form as previously described. We begin by looking at a re-working of Example 3.1.4.

Example 4.3.1. The symmetric group on 3 points, S3 ∼= C3 oC2 has

property B and is a group of the form described in Theorem 4.1.4.

Proof. From Example 3.1.4 we know that S3 has property B and so it re-

mains to show it is of the desired form. TakeV ={0,1,2}to be the additive group of the fieldF3 and H={1,2} the multiplicative group ofF3. Clearly

V ∼= C3 and H ∼= C2. Using the homomorphism φ :H → Aut(V), as we

constructed, ifv∈V then 1φ=α1:v7→v and 2φ=α2 :v7→2v under the

field multiplication fromF3. Forming the semidirect product G=V oφH,

under the action ofφ, and our semigroup multiplication, we observe thatG

is not abelian and so we can deduce it must be be isomorphic toS3.

We now look at how a class of groups all fit the forms constructed in the previous sections. In Chapter 3 Proposition 3.1.6 told us that dihedral groups of order 2p, wherepis an odd prime, have propertyB. We now inves- tigate how these dihedral groups relate to the forms described in Theorems 4.1.4 and 4.2.1.

Proposition 4.3.2. Dihedral groups of order 2p for some odd primep are of the form described in Theorem 4.1.4 being isomorphic to CpoC2.

Proof. It is well known that the dihedral groupDp is isomorphic toCpoC2

where the cyclic group of order two acts by inversion. Also note thatCp is

unique subgroup of order two embedded in the multiplicative group ofFp is

the same as the inversion action of the C2. This is because when p is odd

the element of order two in the multiplicative group of the field F?p is −1.

Now (−1)2= 1 and sincep is not two −1 is not equal to 1 modulo p. Note that, in the notation of our constuction,v((−1)θ) =−v is the inverse ofv

in Fp viewed as an additive group. Thus Dp has the form as described in

Theorem 4.1.4.

Following on from Proposition 3.1.6 we explained that a dihedral group of order 2pnhas Frattini quotient isomorphic toDpand thus by Lemma 3.1.1

Dpn has property B. Therefore we can see thatD_pn satisfies the hypothesis of Theorem 4.2.1.

Proposition 4.3.3. Dihedral groups of order 2pn_{, for some odd prime p,}

are of the form described in Theorem 4.2.1.

Proof. From the proof of Corollary 3.1.7 we saw that the quotient of Dpn by its Frattini subgroup is isomorphic to Dp which from above is CpoC2

and is constructed via multiplication in the field Fp. ThusDpn satisfies the hypothesis of Theorem 4.2.1.

Chapter 5

Classifying Groups with the

Basis Property

A group G is said to have the basis property if all subgroups of G have property B. In this chapter we provide some examples of groups with the basis property and then establish some results that hold for all groups with the basis property. We finish by providing a classification of all groups with the basis property and showing how this links in with the matroid groups classified by Scapellato and Verardi [10].

5.1

An Introduction to Groups with the Basis Prop-

erty

We know that a p-group has property B from our previous work. Since a subgroup of a p-group is itself a p-group we can conclude that all p- groups have the basis property. In fact we can generalise this slightly to

say that any group with propertyB and onlyp-groups as subgroups has the basis property. As a consequence, the smallest non-p-group with the basis property is the symmetric group on 3 points. As we have shown previously this has property B and subgroups isomorphic to C3, C2 and the trivial

subgroup. We now provide an example of a class of groups that have the basis property.

Example 5.1.1. If p is a prime then the dihedral group of order 2pn,Dpn,

has the basis property.

Proof. IfDpn is a p-group then it has the basis property as observed above. So assume thatDpn is not ap-group, i.e. thatp6= 2. ThusD_pn has the form

P ohbi where P is a p-group and hbi is a subgroup of order 2. Let H be

a subgroup of Dpn. First note that H∩P is normal in H and isomorphic to a cyclic group of order of a power of p. Now let π be the mapping from

Dpn to hbi. The kernel of this mapping is P and so Hπ is either trivial or

hbi. IfHπ = 1 thenH=H∩P and so is a cyclicp-group. So letHπ=hbi. NowH has a Sylow 2-subgroup so lethbe a non-trivial element ofHin this Sylow 2-subgroup. Since Dpn is not a p-group then its Sylow 2-subgroup is hbi and so hπ = b. Thus H = (H∩P)hhi = (H∩P)ohhi and h acts

by inversion. Therefore H is isomorphic to a dihedral group of order 2pm

(m≤n) orH is isomorphic toC2 ifH∩P is trivial. Thus any subgroup of

a dihedral group of order 2pn is either a smaller dihedral group of the same form or cyclic of prime-power order. HenceDpn has the basis property.

Of course not all groups with propertyB have the basis property.

the natural map from C9 into C3 and the mapping of C3 into the unique

subgroup of order three intoAut(C2×C2) =S3. ThenGhas propertyB but

not the basis property.

Proof. Note thatφis the composite of the natural map fromC9 intoC3and

the homomorphism that occurs in our construction via field multiplication, specifically the homomorphism C3 7→ Aut(C2×C2) coming from our con-

struction via the multiplication in the fieldF4. Then kerφis isomorphic to

the cyclic group of order 3 and so kerφis the unique subgroup of order 3 in

G. Now letM be a maximal subgroup ofGthat does not contain kerφ. As

M is maximal inGthenGis in fact equal toMkerφ, with the intersection ofM and the kernel trivial, as kerφis a minimal normal subgroup. ThusG

is equal to kerφoM, but this contradicts kerφbeing the unique subgroup

of order 3. This follows asM has a subgroup of order 3 by Sylow’s Theorem. Thus kerφis contained in every maximal subgroup ofGand so is contained in the Frattini subgroup of G. However, the quotient of Gby kerφ is con- structed by field multiplication in F4 and has trivial Frattini subgroup by

Theorem 4.2.1 part (iii); thus G/Φ (G) has property B, and so by Lemma 3.1.1 so doesG. HoweverGdoes not have the basis property, as it contains the subgroupC2×C2×kerφisomorphic to C2×C2×C3.