The Mac Lane method of construction and classification of extensions of cyclic groups by their automorphism groups

M.Kuźma, Institute of Physics Higher Pedagogical School, Rejtana 16a 35–310 Rzeszów, Poland

B.Lulek, Institute of Physics A. Mickiewicz University, Matejki 48/49 60–769 Poznań, Poland

Abstract

The Mac Lane method has been applied to the construction of the second cohomology group in the extension C_{12}×Aut C_{12}. The method simplifies significantly the difficult problem of construction of nonequivalent extensions and allows to investigate their structure.

1 Introduction

Extensions of groups have many applications in physics. Space groups in crystallography being extensions of a three-dimensional translation group by a point group is a classical example of such an application. Many other physical phenomena and theorems can be described in the formalism of extension of groups, too. From this it follows that it is necessary to search methods of obtaining the extensions and investigate their structures.

In this paper we present the application of the Mac Lane method [1,2] to the construction of extensions of finite cyclic groups by a group of its automorphisms. Physical motivation for the investigation of such extensions arises from group-theoretic description of properties of line polymers whose structure is described by a line group [3]. In this case one-dimensional translations form a cyclic group whereas a group of automorphisms, according to the Weyl recipe [4], describes the inner symmetry of a system.

The Mac Lane method allows us to obtain all nonequivalent extensions expressed by a factor system. Calculations were performed for the cyclic group C_{12} having four automorphisms forming a group D_{2}.

2 Nonequivalent extensions of groups

A group G is an extension of the ”passive” group T by an ”active” group Q under a given operator action Δ if it has a normal subgroup T′ ◁ G isomorphic with T and if the quotient group G/T′ is isomorphic with Q.

The groups G, T, Q form an exact sequence:

0 ⟶ T ⟶^{κ} G ⟶^{ω} Q ⟶ 1.  (1)

Denoting the elements of the extension G by ⟨t, q⟩, t ∈ T, q ∈ Q, the multiplication rule in the group G has the form:

⟨t, q⟩ ⟨t′, q′⟩ = ⟨t + qt′ + m(q, q′), qq′⟩ .  (2)

The factors m(q, q′) form the so-called factor system, which fully characterises an extension G.

An extension is described for a given operator action Δ : Q → Aut T of the active group Q on a passive group T:

Δ(q) = ( \begin{matrix}t \\ qt \end{matrix} ) q ∈ Q, t ∈ T.  (3)

Two extensions G and G′ (for the same groups Q and T) are equivalent if there exists an isomorphism χ : G → G′ such that the diagram which represents sequences of these extensions is commutative i.e. χ ∘ κ = κ′ and ω′ ∘ χ = ω

\begin{aligned}0 & ⟶ & T & ⟶^{κ} & G & ⟶^{ω} & Q & ⟶ & 1 \\ & & ↓ id T & & ↓ χ & & ↓ id Q \\ 0 & ⟶ & T′ & ⟶^{κ′} & G′ & ⟶^{ω′} & Q & ⟶ & 1 \end{aligned}  (4)

The equivalency of extensions can also be defined basing on a factor system. Namely two extensions are equivalent if their factor systems differ from each other by a twocoboundary δc

m′ = m + δc.  (5)

3 Second group of cohomology

For given groups Q and T one can obtain many extensions. But not all of them differ. Some of them are equivalent. The number of nonequivalent extensions is given by the second cohomology group H_{Δ}^{2}(Q, T) [5,6]:

H_{Δ}^{2}(Q, T) = Z_{Δ}^{2}(Q, T)/B_{Δ}^{2}(Q, T),  (6)

where Z_{Δ}^{2}(Q, T) is the group of all twococycles, while B_{Δ}^{2}(Q, T) is the group of all twocoboundaries. Groups Z_{Δ}^{2}(Q, T) and B_{Δ}^{2}(Q, T) are subgroups of all twocochains C_{Δ}^{2}(Q, T). The order of this group is great:

|C_{Δ}^{2}(Q, T)| = |T |^{(|Q|^{2})}.  (7)

This order increases when the orders of the groups T and Q are increased (combinatorial explosion) and e.g. for such small groups Q and T as |Q| = 3, |T | = 4 the order of the group of twococycles is equal to 262144.

4 The Mac Lane method

Mac Lane’s theorem helps to construct the second cohomology group and enables us to inspect some features of the structure of extension. The essence of the method is in a theorem that a second cohomology group is isomorphic with a quotient group for another exact sequence involving free groups [1]:

[figure: 0; 1; T; R; G; F; Q; 1; κ; i; ω; M; γ; φ]  (8)

where F, R are free groups with alphabet ⟨X⟩ and ⟨Y ⟩ respectively.

The group F is generated freely from a set A ⊂ Q of generators of an active group Q. The free group R ◁ F is a quotient group of F and its alphabet is formed using the Nielson–Schreier theorem [5]:

Y = {sxβ(sx)^{−1} | x ∈ X, s ∈ S}, S = {f_{q} | q ∈ Q},  (9)

where S is the Nielson–Schreier set, f_{q} — representatives of cosets of F. In diagram (8) φ denotes an operator homomorphisms from R to T and γ denotes a crossed homomorphisms from F to T.

Denoting the set of all operator and crossed homomorphisms by Z_{Δ∘M}^{1}(F, T) we can derive the second cohomology group from an isomorphism:

H_{Δ}^{2}(Q, T) ≅ Hom_{F} (R, T)/Z_{Δ∘M}^{1}(F, T)|_{R},  (10)

where Z_{Δ∘M}^{1}(F, T) is restricted to R.

One can construct a group Hom_{F} (R, T) from a manifold of all mappings from the alphabet Y of the group R on to group T

Hom_{F} (R, T) = T^{Y} = {φ | Y → T }.  (11)

Then we have to select from this manifold T^{Y} the submanifold of all operator homomorphisms i.e. such mappings, which intertwine the action Ξ : F → Aut R and Δ : Q → Aut T. They form conditions for the operator homomorphism:

φ(xyx^{−1}) = M(x)φ(y) x ∈ X, y ∈ Y.  (12)

The group Z_{Δ∘M}^{1}(F, T) of crossed homomorphisms is derived from other rules:

γ(f_{1}f_{2}) = γ(f_{1}) + M(f_{1})γ(f_{2}), f_{1}f_{2} ∈ F^{2}.  (13)

5 Extensions of C_{12} × Aut C_{12}

The presented method has been applied to the construction of all nonequivalent extensions of groups C_{12} ×Aut C_{12}, where the translation group T is a cyclic group C_{12} = {j | j = 1, 2, …, 12}

Table 1: The group Aut C_{12}

\begin{matrix}τ\j & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ τ_{1} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ τ_{5} & 5 & 10 & 3 & 8 & 1 & 6 & 11 & 4 & 9 & 2 & 7 & 12 \\ τ_{7} & 7 & 2 & 9 & 4 & 11 & 6 & 1 & 8 & 3 & 10 & 5 & 12 \\ τ_{11} & 11 & 10 & 9 & 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1 & 12 \end{matrix}

Table 2: Operator actions Δ : Q → Aut C_{12}

τ\Δ Δ_{1} Δ_{2} Δ_{3} Δ_{4} Δ_{5} Δ_{6} Δ_{7} Δ_{8} Δ_{9} Δ_{10} Δ_{11} Δ_{12} Δ_{13} Δ_{14} Δ_{15} Δ_{16}
τ_{1} τ_{1} τ_{1} τ_{1} τ_{1} τ_{1} τ_{1} τ_{1} τ_{1} τ_{1} τ_{1} τ_{1} τ_{1} τ_{1} τ_{1} τ_{1} τ_{1}
τ_{5} τ_{1} τ_{1} τ_{1} τ_{1} τ_{5} τ_{5} τ_{5} τ_{5} τ_{7} τ_{7} τ_{7} τ_{7} τ_{11} τ_{11} τ_{11} τ_{11}
τ_{7} τ_{1} τ_{5} τ_{7} τ_{11} τ_{1} τ_{5} τ_{7} τ_{11} τ_{1} τ_{7} τ_{5} τ_{11} τ_{1} τ_{5} τ_{7} τ_{11}
τ_{11} τ_{1} τ_{5} τ_{7} τ_{11} τ_{5} τ_{1} τ_{11} τ_{7} τ_{7} τ_{1} τ_{11} τ_{5} τ_{11} τ_{7} τ_{5} τ_{1}

and the point group Q is the group of automorphisms Aut C_{12} = {τ_{r} | r = 1, 5, 7, 11}. The latter one has four elements τ_{r} described by relation τ_{r}j = rj mod 12, j ∈ C_{12}. This group (cf also Table 1) is isomorphic to the group D_{2}. All possible operator actions Δ : Q → Aut C_{12} have been listed in Table 2.

We express nonequivalent extensions for the operator action Δ_{7} (Table 2) in terms of a factor system defined by

m(q_{1}, q_{2}) = φ(ϱ(q_{1}, q_{2})), (q_{1}, q_{2}) ∈ Q^{2}.  (14)

The function ϱ(q_{1}, q_{2}) in (14) is described by a product

f_{q_{1}}f_{q_{2}} = ϱ(q_{1}, q_{2})f_{q_{1}q_{2}},  (15)

where f_{q} is a representative of the coset in the decomposition

F = ⋃_{q∈Q} Rf_{q},  (16)

and R is the kernel of the epimorphism M : F → Q. The group D_{2} has an alphabet X = {x_{1}, x_{2}} and the Schreier set consists of four elements S = {e_{F} , x_{1}, x_{2}, x_{1}x_{2}}. This set determines the factor system ϱ : Q×Q → R by (15). For our case the factor system ϱ(q_{1}, q_{2}) has been presented in Table 3.

The alphabet Y of the subgroup R can be identify with the set of all non-trivial elements of the second and third columns of Table 3. Thus we have

Y = {y_{1} = x_{1}^{2}, y_{2} = x_{2}^{2}, y_{3} = x_{2}x_{1}x_{2}^{−1}x_{1}^{−1}, y_{4} = x_{1}x_{2}x_{1}x_{2}^{−1}, y_{5} = x_{1}x_{2}^{2}x_{1}^{−1}.  (17)

The factor system ϱ : Q × Q → R, expressed in terms of the alphabet Y is presented in Table 4. Having the alphabets X and Y we can construct conditions for operator homomorphisms (12)

Table 3: The factor system ϱ : Q × Q → R in the alphabet X

e_{F}  x_{1}  x_{2}  x_{1}x_{2}
e_{F} e_{F}  e_{F}  e_{F}  e_{F}
x_{1} e_{F}  x_{1}^{2}  e_{F}  x_{1}^{2}
x_{2} e_{F} x_{2}x_{1}x_{2}^{−1}x_{1}^{−1}  x_{2}^{2}  x_{2}x_{1}x_{2}x_{1}^{−1}
x_{1}x_{2} e_{F} x_{1}x_{2}x_{1}x_{2}^{−1} x_{1}x_{2}^{2}x_{1}^{−1} x_{1}x_{2}x_{1}x_{2}

Table 4: The factor system ϱ : Q × Q → R in the alphabet Y

E u_{x} u_{y} u_{z}
E e_{F} e_{F} e_{F} e_{F}
u_{x} e_{F} y_{1} e_{F} y_{1}
u_{y} e_{F} y_{3} y_{2} y_{3}y_{5}
u_{z} e_{F} y_{4} y_{5} y_{4}y_{2}

(see Table 5). Both, operator and crossed homomorphisms, forming groups Hom_{F} (R, C_{12}) and Z_{Δ∘M}^{1}, are collected in Table 6 and 7, respectively.

According to formula (6) we construct cosets determined by Z_{Δ∘M}^{1}(F, T) in Hom_{F} (R, T) (see Table 8) forming the second cohomology group H_{Δ_{7}}^{2}(Q, T). This group has eight elements. It yields that we have eight nonequivalent extensions C_{12} × Aut C_{12} (under operator action Δ_{7}). Such extensions are listed in Table 9 in the form of factor systems m : Q × Q → C_{12} for the coset representatives chosen as in Table 8.

6 Conclusions

The second cohomology group for an extension C_{12} × Aut C_{12} is of the order 8 (under the operator action Δ_{7} described in Table 2). Each element of this group m ∈ H^{2}(D_{2}, C_{12}) forms the factor system (Table 9) determined by the Seitz formula (2) and gives an nonequivalent

Table 5: Conditions for operator homomorphisms φ : R → T

x_{1}  x_{2}
y_{1}  φ(y_{1}) = 5φ(y_{1})  φ(y_{3}) + φ(y_{4}) = 7φ(y_{1})
y_{2}  φ(y_{5}) = 5φ(y_{2})  φ(y_{2}) = 7φ(y_{2})
y_{3} φ(y_{4}) − φ(y_{1}) = 5φ(y_{3})  φ(y_{2}) − φ(y_{5}) − φ(y_{3}) = 7φ(y_{3})
y_{4} φ(y_{1}) + φ(y_{3}) = 5φ(y_{4}) φ(y_{1}) − φ(y_{2}) + φ(y_{3}) + φ(y_{5}) = 7φ(y_{4})
y_{5}  φ(y_{2}) = 5φ(y_{5})  φ(y_{5}) = 7φ(y_{5})

Table 6: The group Hom_{F} (R, C_{12}) of operator homomorphisms (for operation action Δ_{7})

y_{1} y_{2} y_{3} y_{4} y_{5}  y_{1} y_{2} y_{3} y_{4} y_{5}
φ_{1} 3 2 5 4 10 φ_{25} 9 2 5 10 10
φ_{2} 3 2 11 10 10 φ_{26} 9 2 11 4 10
φ_{3} 3 4 1 8 8 φ_{27} 9 4 1 2 8
φ_{4} 3 4 7 2 8 φ_{28} 9 4 7 8 8
φ_{5} 3 6 3 6 6 φ_{29} 9 6 3 12 6
φ_{6} 3 6 9 12 6 φ_{30} 9 6 9 6 6
φ_{7} 3 8 5 4 4 φ_{31} 9 8 5 10 4
φ_{8} 3 8 11 10 4 φ_{32} 9 8 11 4 4
φ_{9} 3 10 1 8 2 φ_{33} 9 10 1 2 2
φ_{10} 3 10 7 2 2 φ_{34} 9 10 7 8 2
φ_{11} 3 12 3 6 12 φ_{35} 9 12 3 12 12
φ_{12} 3 12 9 12 12 φ_{36} 9 12 9 6 12
φ_{13} 6 2 2 4 10 φ_{37} 12 2 2 10 10
φ_{14} 6 2 8 10 10 φ_{38} 12 2 8 4 10
φ_{15} 6 4 4 2 8 φ_{39} 12 4 4 8 8
φ_{16} 6 4 10 8 8 φ_{40} 12 4 10 2 8
φ_{17} 6 6 6 12 6 φ_{41} 12 6 6 6 6
φ_{18} 6 6 12 6 6 φ_{42} 12 6 12 12 6
φ_{19} 6 8 2 4 4 φ_{43} 12 8 2 10 4
φ_{20} 6 8 8 10 4 φ_{44} 12 8 8 4 4
φ_{21} 6 10 4 2 2 φ_{45} 12 10 4 8 2
φ_{22} 6 10 10 8 2 φ_{46} 12 10 10 2 2
φ_{23} 6 12 6 12 12 φ_{47} 12 12 6 6 12
φ_{24} 6 12 12 6 12 φ_{48} 12 12 12 12 12

Table 7: The group Z_{Δ∘M}^{1} of crossed homomorphisms (for operation action Δ_{7})

y_{1} y_{2} y_{3} y_{4} y_{5}
γ_{1} 6 8 2 4 4
γ_{2} 6 4 10 8 8
γ_{3} 6 12 6 12 12
γ_{4} 12 8 8 4 4
γ_{5} 12 4 4 8 8
γ_{6} 12 12 12 12 12

Table 8: Coset representatives Hom_{F} (R, T)/Z_{Δ∘M}^{1}(F, T) (for operation action Δ_{7})

y_{1} y_{2} y_{3} y_{4} y_{5}
r_{0} 12 12 12 12 12
r_{1} 9 6 9 6 6
r_{2} 3 6 9 12 6
r_{3} 3 12 3 6 12
r_{4} 9 12 3 12 12
r_{5} 6 6 12 6 6
r_{6} 6 6 6 12 6
r_{7} 6 12 12 6 12

Table 9: Factor systems m : Q × Q → C_{12} (for operation action Δ_{7})

\begin{matrix}r_{0} = & ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} ) & r_{1} = & ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 9 & 6 & 3 \\ 0 & 9 & 6 & 3 \\ 0 & 6 & 6 & 0 \end{matrix} ) \\ r_{2} = & ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 3 & 0 & 3 \\ 0 & 9 & 6 & 3 \\ 0 & 0 & 6 & 6 \end{matrix} ) & r_{3} = & ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 3 & 0 & 3 \\ 0 & 3 & 0 & 3 \\ 0 & 6 & 0 & 6 \end{matrix} ) \\ r_{4} = & ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 9 & 0 & 9 \\ 0 & 3 & 0 & 3 \\ 0 & 0 & 0 & 0 \end{matrix} ) & r_{5} = & ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 6 & 0 & 6 \\ 0 & 0 & 6 & 6 \\ 0 & 6 & 6 & 0 \end{matrix} ) \\ r_{6} = & ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 6 & 0 & 6 \\ 0 & 6 & 6 & 0 \\ 0 & 0 & 6 & 6 \end{matrix} ) & r_{7} = & ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 6 & 0 & 6 \\ 0 & 0 & 0 & 0 \\ 0 & 6 & 0 & 6 \end{matrix} ) \end{matrix}

extension. However, the factor system depends on a choice of coset representatives listed in table 8. This choice corresponds to the gauge transformation [2], which is connected with equivalent extensions.

References

[1] S. Mac Lane, Ann. Math. 50, 736 (1949).

[2] T. Lulek, Acta Phys. Pol. 82, 377 (1992); T. Lulek and R. Chatterjee, ibid. 395 (1992).

[3] M. Vujicič, I.B. Bozovič and F. Herbut, J. Phys. A10, 1271 (1977).

[4] H. Weyl, Symmetry, Princeton UP, New Jersey 1952.

[5] M.I. Kargapolov and Y.I. Merzlakov, Principles of Group Theory, Nauka, Moscow 1982 (in Russian).

[6] A.G. Kurosh, Group Theory, Nauka, Moscow 1967 (in Russian).