/compile/output.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 8, No. 3, 2015, 375-388

ISSN 1307-5543 – www.ejpam.com

Analyzing Group Theoretical Properties of Groups with Order

24 by GAP

Ahmet Faruk Aslan, Alper Odabaş∗

Department of Mathematics - Computer, Science and Art Faculty, Eski̧sehir Osmangazi University,

Eski̧sehir, Turkey

Abstract. We will present our implementation fgroup.gi and present subgroup lattice of some groups

of order 24. We also will construct tables containing those groups basic properties and generators ele-

ments of Frattini and Fitting subgroups.

2010 Mathematics Subject Classifications: 20D05, 20D25, 20D30

Key Words and Phrases: Subgroup lattice, frattini subgroup, fitting subgroup.

1. Introduction

The link between group theory and GAP is constructed by Cayley’s theorem which states

that every group is isomorphic to a subgroup of a permutation group. If G is a group with

order n, then we can define an isomorphism from G into the symmetric group Sn by Cayley’s

theorem. GAP generates all finite order groups using the symmetric group operations, and

gives concrete examples to the abstract notions of algebra, in other words, it visualizes these

abstract notions. Some applications of GAP to abstract algebra can be found in [1, 3].

Since most of the cohomological invariants are (directly) related to nilpotency. It is well

known that the Frattini and Fitting subgroups are related to nilpotency. In the sense of this

relation we compute the generator elements of these special subgroups. It is one of the fun-

damental property of algebraic structures such as groups, algebras, algebraic models etc.

The goals of this paper are:

* Presenting our implementation fgroup.gi

* Constructing tables consisting of some basic properties and generator elements of Fitting,

Frattini subgroups of a group of order 24.

* Constructing the subgroup lattices of these groups.

∗Corresponding author.

Email addresses: afaslan@ogu.edu.tr (A. Aslan), aodabas@ogu.edu.tr (A. Odabaş)

http://www.ejpam.com 375 c© 2015 EJPAM All rights reserved.



A. Aslan, A. Odabaş / Eur. J. Pure Appl. Math, 8 (2015), 375-388 376

2. Preliminaries

In GAP programming, the following functions are frequently used for constructing groups

and obtain some of their properties.

Group
GeneratorsOfGroup
ConjugateGroup
IsGroup
IsCyclic
IsAbelian
Elements

Example 1. The cyclic group C4 and its properties can be obtained by the following GAP com-

mands,

gap> G:=Group((1,2,3,4));
Group([ (1,2,3,4) ])
gap> Size(G);
4
gap> IsCyclic(G);
true
gap> IsAbelian(G);
true

The function Subgroup() is used for obtain subgroups. Below we give the useful GAP

commands related with subgroups and cosets.

Subgroup
Index
IsSubgroup
IsNormal
ConjugateSubgroup
RightCoset
IsRightCoset
ConjugacyClasses

Example 2. The following coding can be used for constructing subgroups.

gap> G:=Group((1,2,3,4),(1,2,3));
Group([ (1,2,3,4), (1,2,3) ])
gap> Subgroup(G,[(1,2,3,4)]);
Group([ (1,2,3,4) ])
gap> H:=Subgroup(G,[(1,2,3,4)]);
Group([ (1,2,3,4) ])



A. Aslan, A. Odabaş / Eur. J. Pure Appl. Math, 8 (2015), 375-388 377

Also the conjugacy classes obtained by the function ConjugacyClasses() and the func-

tion NrConjugacyClasses() is used for finding the number of conjugacy classes of the

group.

gap> G:=Group((1,2,3,4,5));
Group([ (1,2,3,4,5) ])
gap> ConjugacyClasses(G);
[()^G,(1,2,3,4,5)^G,(1,3,5,2,4)^G,(1,4,2,5,3)^G,
(1,5,4,3,2)^G ]
gap> NrConjugacyClasses(G);
5

The functions FactorGroup() and FactorGroupNC() are used for constructing quotient

groups and their enumerations. Here we will construct all quotient groups of a group by using

second author’s implementation in [3] which we called as fgroup.gi.

gap> Read("fgroup.gi");
gap> G:=Group((1,2,3),(1,3));
Group([ (1,2,3), (1,3) ])
gap> IsAbelian(G);
false
gap> IsCyclic(G);
false
gap> Fgroup(G);
Group( [ (1,2,3), (1,3) ] ) has 3 factor groups.
These :
[Group([ ]), Group([ f1 ]), Group([ (1,2,3), (1,3) ])]

Finally, in GAP, a homomorphism between groups is defined as follows

gap> G:=Group((1,2,3)(4,5));
Group([ (1,2,3)(4,5) ])
gap> genG:=GeneratorsOfGroup(G);
[ (1,2,3)(4,5) ]
gap> H:=Group((1,2)(3,4));
Group([ (1,2)(3,4) ])
gap> hom:=GroupHomomorphismByImages(G,H,genG,[(1,2)]);
[ (1,2,3)(4,5) ] -> [ (1,2) ]
gap> IsGroupHomomorphism(hom);
true

It is well known that for a given group G and its normal subgroup H, we have G/H is

abelian if and only if [G, G] ⊆ H, where [G, G] is the commutator subgroup of G. So, [G, G] is

the smallest normal subgroup which makes the quotient abelian. Consequently, we have the

following example.

Example 3. We will find the commutator subgroup of S8 and look its normality and show that

S8/[S8,S8] is the largest quotient group in the abelian qoutients of S8.



A. Aslan, A. Odabaş / Eur. J. Pure Appl. Math, 8 (2015), 375-388 378

gap> G:=SymmetricGroup(8);
Sym( [ 1 .. 8 ] )
gap> Size(G);
40320
gap> K:=DerivedSubgroup(G);
Group([ (1,3,2), (1,4,3), (1,4,5), (1,5,6),
(1,4,3,6,7), (1,6,8)(3,4)(5,7) ])
gap> IsNormal(G,K);
true
gap> F:=FactorGroup(G,K);
Group([ f1 ])
gap> IsAbelian(F);
true
gap> Fgroup(G);
SymmetricGroup( [ 1 .. 8 ] ) has 3 factor groups.
These :
[ Sym( [ 1 .. 8 ] ), Group([ f1 ]), Group([ ]) ]

Definition 1. The largest nilpotent normal subgroup of a group G, is called the Fitting subgroup,

denoted F(G). The Frattini subgroup of a group G, denotedΦ(G), is the intersection of all maximal

subgroups of G. Of course, Φ(G) is characteristic, and hence normal in G, and it is nilpotent. It

follows that for any finite group G, we have Φ(G)≤ F(G).

We refer [4] and [2] for details about the Fitting and Frattini subgroups.

Theorem 1 (Frattini Argument). Let NG(P) is normaliser of P in G, N Ã G and suppose that

P ∈ S ylp(N). Then G = NG(P)N.

Theorem 2. Let Φ(G)≤ N Ã G and suppose that N/Φ(G) is nilpotent. Then N is nilpotent.

Proof. To show that N is nilpotent, we prove that each of its Sylow subgroups is normal.

For this purpose, we let P ∈ S ylp(N) and we note that PΦ(G)/Φ(G) is a Sylow p-sugroup of

N/Φ(G). But N/Φ(G) is assumed to be nilpotent, and thus its Sylow subgroups are normal

and we have PΦ(G)/Φ(G) Ã N/Φ(G). Thus, in fact, PΦ(G)/Φ(G) is actually characteristic in

N/Φ(G) and since N/Φ(G) Ã G/Φ(G), we deduce that PΦ(G)/Φ(G) Ã G/Φ(G). This gives

PΦ(G)Ã G.

Since P is Sylow in N , it is also Sylow in PΦ(G). Since the latter subgroup is normal in

G, we can apply the Frattini argument to deduce that NG(P)PΦ(G) = G. Since P ≤ NG(P),

however, this yields NG(P)Φ(G) = G. It follows from this that NG(P) = G (Otherwise, NG(P)

would be contained in some maximal subgroup of G, which also contains Φ(G), and this would

contradict the fact that NG(P)Φ(G) = G). We now have NG(P) = G, and thus P Ã G. In

particular, P Ã N as desired.

Example 4. The Fitting and Frattini subgroups of the Dihedral group with order 28 can be ob-

tained by GAP as follows:



A. Aslan, A. Odabaş / Eur. J. Pure Appl. Math, 8 (2015), 375-388 379

gap> D:=DihedralGroup(28);
<pc group of size 28 with 3 generators>
gap> FittingSubgroup(D);
Group([ f2*f3^3, f3^5 ])
gap> F1:=FittingSubgroup(D);
Group([ f2*f3^3, f3^5 ])
gap> Size(F);
14
gap> F2:=FrattiniSubgroup(D);
Group([ ])
gap> G:=DihedralGroup(28);
gap> Ns := NormalSubgroups(D);
[ Group([ ]), Group([ f2*f3^3 ]), Group([ f3 ]),

Group([ f1*f2, f3 ]), Group([ f1, f3 ]), Group([ f2, f3 ]),
<pc group of size 28 with 3 generators> ]

gap> N := Ns[2];;
gap> FF:=FactorGroup(N,F);
Group([ f2*f3^3 ])
gap> IsNilpotentGroup(FF);
true
gap> IsNilpotentGroup(N);
true

2.1. Constructing Subgroup Lattices by GAP

Finding subgroups and classification of group properties of a group has significant impor-

tance for analyzing its relations to compare with other groups.

In this section we will look how GAP is used for classifying subgroups. For example sub-

group lattice of Klein 4-group is as shown in Figure 1 [5].

Figure 1: Subgroup Lattice of Klein 4-group.



A. Aslan, A. Odabaş / Eur. J. Pure Appl. Math, 8 (2015), 375-388 380

This means that the group has five subgroups where three of them are trivial and others are

2/1 type, i.e., cyclic group of order 2. These subgroups can be find by using GAP as follows:

gap> k4:=Group((1,2),(3,4));
Group([ (1,2), (3,4) ])
gap> l:=LatticeSubgroups(k4);
<subgroup lattice of Group([ (1,2), (3,4) ]),
5 classes, 5 subgroups>
gap> IsAbelian(k4);
true
gap> ConjugacyClassesSubgroups(l);
[ Group( () )^G, Group( [ (3,4) ] )^G, Group( [ (1,2) ] )^G,

Group( [ (1,2)(3,4) ] )^G, Group( [ (3,4), (1,2) ] )^G ]
gap> M:=MaximalSubgroupsLattice(l);
[ [ ], [ [ 1, 1 ] ], [ [ 1, 1 ] ], [ [ 1, 1 ] ],

[ [ 4, 1 ], [ 3, 1 ], [ 2, 1 ] ] ]
gap> M[5];
[ [ 4, 1 ], [ 3, 1 ], [ 2, 1 ] ]
gap> u1:=Representative(ConjugacyClassesSubgroups(l)[5]);
Group([ (3,4), (1,2) ])
gap> u2:=ClassElementLattice(ConjugacyClassesSubgroups(l)[4],1);;
gap> u3:=ClassElementLattice(ConjugacyClassesSubgroups(l)[3],1);;
gap> u4:=ClassElementLattice(ConjugacyClassesSubgroups(l)[2],1);;
gap> IsSubgroup(u1,u2);IsSubgroup(u1,u3);IsSubgroup(u1,u4);
true
true
true

2.2. Examples

In this section, up to GAP order, we will give tables containing generators of (Frattini,

Fitting) subgroups, abelianess property of 14 different groups of order 24. In all cases, the

table headings are as defined in Table 1. Also, we give the subgroup lattices near the tables.

Table 1: Group & Generator Properties

Group T Type of Group NS Number of Subgroups

Group N Name of Group NNS Number of Normal Subgroups

GAP T Type of GAP FiG Generators of Fitting Subgroup

GAP N GAP Name of Group FrG Generators of Frattini Subgroup

Deg. Degree of Group Gen. Generators of Group

Ab. Group is Abelian



A. Aslan, A. Odabaş / Eur. J. Pure Appl. Math, 8 (2015), 375-388 381

Group T 24/2

Group N C2 × C12

GAP T 24/2

GAP N c12c2

Deg. 24

Ab. Abelian

NS 16

NNS 16

FiG (7,9,8), (3,5)(4,6), (3,6,5,4), (1,2)

FrG (3,5)(4,6)

Gen. (1,2), (3,4,5,6)(7,8,9)

Table 2: Table of C2 × C12 Figure 2: Subgroup lattice of C2 × C12

Group T 24/3

Group N C6 × C2
2

GAP T 24/1

GAP N c6k4

Deg. 24

Ab. Abelian

NS 32

NNS 32

FiG (8,9), (6,7), (4,5), (1,2,3)

FrG ()

Gen. (1,2,3)(4,5), (6,7), (8,9)

Table 3: Table of C6 × C2
2 Figure 3: Subgroup lattice of C6 × C2

2



A. Aslan, A. Odabaş / Eur. J. Pure Appl. Math, 8 (2015), 375-388 382

Group T 24/4

Group N D6 × C2

GAP T 24/6

GAP N d12c2

Deg. 24

Ab. Not Abelian

NS 54

NNS 21

FiG (6,7), (4,5), (1,3,2)

FrG ()

Gen. (1,2,3)(4,5), (2,3), (6,7)

Table 4: Table of D6 × C2 Figure 4: Subgroup lattice of D6 × C2

Group T 24/5

Group N A4 × C2

GAP T 24/10

GAP N a4c2

Deg. 24

Ab. Not Abelian

NS 26

NNS 6

FiG (5,6), (1,2)(3,4), (1,3)(2,4)

FrG ()

Gen. (1,2,3), (2,3,4), (5,6)

Table 5: Table of A4 × C2 Figure 5: Subgroup lattice of A4 × C2



A. Aslan, A. Odabaş / Eur. J. Pure Appl. Math, 8 (2015), 375-388 383

Group T 24/6

Group N Q6 × C2

GAP T 24/8

GAP N q12c2

Deg. 24

Ab. Not Abelian

NS 22

NNS 13

FiG (8,9), (5,7,6), (1,2)(3,4)

FrG (1,2)(3,4)

Gen. (1,2)(3,4)(5,6,7),

(1,3,2,4)(6,7), (8,9)

Table 6: Table of Q6 × C2 Figure 6: Subgroup lattice of Q6 × C2

Group T 24/7

Group N D4 × C3

GAP T 24/4

GAP N d8c3

Deg. 24

Ab. Not Abelian

NS 20

NNS 12

FiG (5,7,6), (2,4), (1,3)(2,4), (1,43,2)

FrG (1,3)(2,4)

Gen. (1,2,3,4)(5,6,7), (2,4)

Table 7: Table of D4 × C3 Figure 7: Subgroup lattice of D4 × C3



A. Aslan, A. Odabaş / Eur. J. Pure Appl. Math, 8 (2015), 375-388 384

Group T 24/8

Group N Q× C3

GAP T 24/5

GAP N q8c3

Deg. 24

Ab. Not Abelian

NS 12

NNS 12

FiG (9,11,10), (1,2)(3,4)(5,6)(7,8)

(1,6,2,5)(3,8,4,7)

(1,8,2,7)(3,5,4,6)

FrG (1,2)(3,4)(5,6)(7,8)

Gen. (1,5,2,6)(3,7,4,8),

(1,7,2,8)(3,6,4,5), (9,10,11)

Table 8: Table of Q× C3 Figure 8: Subgroup lattice of Q× C3

Group T 24/9

Group N S3 × C4

GAP T 24/7

GAP N s3c4

Deg. 24

Ab. Not Abelian

NS 26

NNS 11

FiG (4,6)(5,7), (4,7,6,5), (1,2,3)

FrG (4,6)(5,7)

Gen. (1,2), (2,3), (4,5,6,7)

Table 9: Table of S3 × C4 Figure 9: Subgroup lattice of S3 × C4



A. Aslan, A. Odabaş / Eur. J. Pure Appl. Math, 8 (2015), 375-388 385

Group T 24/10

Group N D12

GAP T 24/12

GAP N d24

Deg. 24

Ab. Not Abelian

NS 34

NNS 9

FiG (4,6)(5,7), (4,7,6,5), (1,2,3)

FrG (4,6)(5,7)

Gen. (1,2,3)(4,5,6,7), (2,3)(4,7)(5,6)

Table 10: Table of D12 Figure 10: Subgroup lattice of D12

Group T 24/11

Group N Q12

GAP T 24/13

GAP N q24

Deg. 24

Ab. Not Abelian

NS 18

NNS 9

FiG (9,10,11), (1,3)(2,4)(5,7)(6,8),

(1,4,3,2)(5,8,7,6)

FrG (1,3)(2,4)(5,7)(6,8)

Gen. (1,2,3,4)(5,6,7,8)(9,10,11),

(1,5,3,7)(2,8,4,6)(10,11)

Table 11: Table of Q12 Figure 11: Subgroup lattice of Q12



A. Aslan, A. Odabaş / Eur. J. Pure Appl. Math, 8 (2015), 375-388 386

Group T 24/12

Group N S4

GAP T 24/15

GAP N s24

Deg. 24

Ab. Not Abelian

NS 34

NNS 9

FiG (4,6)(5,7), (4,7,6,5), (1,2,3)

FrG (4,6)(5,7)

Gen. (1,2,3)(4,5,6,7), (2,3)(4,7)(5,6)

Table 12: Table of S4 Figure 12: Subgroup lattice of S4

Group T 24/13

Group N SL2(F3)

GAP T 24/14

GAP N sl(2,3)

Deg. 24

Ab. Not Abelian

NS 15

NNS 4

FiG (1,3)(2,4)(5,7)(6,8), (1,4,3,2)

(5,6,7,8), (1,8,3,6)(2,5,4,7)

FrG (1,3)(2,4)(5,7)(6,8)

Gen. (1,2,3,4)(5,8,7,6),

(1,5,3,7)(2,6,4,8),

(2,5,6)(4,7,8)(9,10,11)

Table 13: Table of SL2(F3) Figure 13: Subgroup lattice of SL2(F3)



A. Aslan, A. Odabaş / Eur. J. Pure Appl. Math, 8 (2015), 375-388 387

Group T 24/14

Group N C3 × C8

GAP T 24/9

GAP N c3× c8

Deg. 24

Ab. Not Abelian

NS 10

NNS 7

FiG (4,6,8,10)(5,7,9,11), (4,8)(5,9)

(6,10)(7,11), (1,2,3)

FrG (4,6,8,10)(5,7,9,11), (4,8)(5,9)

(6,10)(7,11)

Gen. (1,2,3), (2,3)(4,5,6,7,8,9,10,11)

Table 14: Table of C3 × C8 Figure 14: Subgroup lattice of C3 × C8

Group T 24/15

Group N D8 × C3

GAP T 24/11

GAP N d8× c3

Deg. 24

Ab. Not Abelian

NS 30

NNS 9

FiG (5,6,7), (2,4), (1,3)(2,4)

FrG (1,3)(2,4)

Gen. (5,6,7), (1,2,3,4)(6,7), (2,4)

Table 15: Table of D8 × C3 Figure 15: Subgroup lattice of D8 × C3



REFERENCES 388

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University. p. 254. (2004).

[4] H. E. Rose. A Course on Finite Groups. Springer, (2000).

[5] A. D. Thomas and G. Wood. Group Tables. Birkhäuser, Boston, p. 248. (1980).