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Finite Group C_2×C_2


The finite group C_2×C_2 is one of the two distinct groups of group order 4. The name of this group derives from the fact that it is a group direct product of two C_2 subgroups. Like the group C_4, C_2×C_2 is an Abelian group. Unlike C_4, however, it is not cyclic.

The abstract group corresponding to C_2×C_2 is called the vierergruppe. Examples of the C_2×C_2 group include the point groups D_2, C_(2h), and C_(2v), and the modulo multiplication groups M_8 and M_(12) (and no other modulo multiplication groups). That M_8, the residue classes prime to 8 given by {1,3,5,7}, are a group of type C_2×C_2 can be shown by verifying that

 1^2=1  3^2=9=1  5^2=25=1  7^2=49=1 (mod 8)
(1)

and

 3·5=15=7  3·7=21=5  5·7=35=3 (mod 8).
(2)

C_2×C_2 is therefore a modulo multiplication group.

FiniteGroupC2C2CycleGraph

The cycle graph is shown above. In addition to satisfying A_i^4=1 for each element A_i, it also satisfies A_i^2=1, where 1 is the identity element.

FiniteGroupC2C2Table

Its multiplication table is illustrated above and enumerated below (Cotton 1990, p. 11).

C_2×C_21ABC
11ABC
AA1CB
BBC1A
CCBA1

Since the group is Abelian, the conjugacy classes are {1}, {A}, {B}, and {C}. Nontrivial proper subgroups of C_2×C_2 are {I,A}, {I,B}, and {I,C}.

Now explicitly consider the elements of the C_(2v) point group.

C_(2v)EC_2sigma_vsigma_v
EEC_2sigma_vsigma_v^'
C_2C_2Esigma_v^'sigma_v
sigma_vsigma_vsigma_v^'EC_2
sigma_v^'sigma_v^'sigma_vC_2E

In terms of the vierergruppe elements

VIV_1V_2V_3
IIV_1V_2V_3
V_1V_1IV_3V_2
V_2V_2V_3IV_1
V_3V_3V_2V_1I

C_2×C_2 has cycle index

 Z(C_2×C_2)=1/2x_1^2+1/2x_2^2.
(3)

A reducible representation using two-dimensional real matrices is

1=[1 0; 0 1]
(4)
A=[-1 0; 0 -1]
(5)
B=[0 1; 1 0]
(6)
C=[0 -1; -1 0].
(7)

Another reducible representation using three-dimensional real matrices can be obtained from the symmetry elements of the D_2 group (1, C_2(z), C_2(y), and C_2(x)) or C_(2v) group (1, C_2, sigma_v, and sigma_v^'). Place the C_2 axis along the z-axis, sigma_v in the x-y plane, and sigma_v^' in the y-z plane.

1=[1 0 0; 0 1 0; 0 0 1]
(8)
A=R_x(pi)=sigma_v=[1 0 0; 0 -1 0; 0 0 1]
(9)
C=R_z(pi)=C_2=[-1 0 0; 0 -1 0; 0 0 1]
(10)
B=R_y(pi)=sigma_v^'=[-1 0 0; 0 1 0; 0 0 1].
(11)

In order to find the irreducible representations, note that the traces are given by chi(1)=3,chi(C_2)=-1, and chi(sigma_v)=chi(sigma_v^')=1. Therefore, there are at least three distinct conjugacy classes. However, we see from the multiplication table that there are actually four conjugacy classes, so group rule 5 requires that there must be four irreducible representations. By group rule 1, we are looking for positive integers which satisfy

 l_1^2+l_2^2+l_3^2+l_4^2=4.
(12)

The only combination which will work is

 l_1=l_2=l_3=l_4=1,
(13)

so there are four one-dimensional representations. Group rule 2 requires that the sum of the squares equal the group order h=4, so each one-dimensional representation must have group character +/-1. Group rule 6 requires that a totally symmetric representation always exists, so we are free to start off with the first representation having all 1s. We then use orthogonality (group rule 3) to build up the other representations. The simplest solution is then given by

C_(2v)1C_2sigma_vsigma_v^'
Gamma_11111
Gamma_21-1-11
Gamma_31-11-1
Gamma_411-1-1

These can be put into a more familiar form by switching Gamma_1 and Gamma_3, giving the character table

C_(2v)1C_2sigma_vsigma_v^'
Gamma_31-11-1
Gamma_21-1-11
Gamma_11111
Gamma_411-1-1

The matrices corresponding to this representation are now

1=[1 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 1]
(14)
C_2=[-1 0 0 0; 0 -1 0 0; 0 0 1 0; 0 0 0 1]
(15)
sigma_v=[1 0 0 0; 0 -1 0 0; 0 0 1 0; 0 0 0 -1]
(16)
sigma_v^'=[-1 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 -1],
(17)

which consist of the previous representation with an additional component. These matrices are now orthogonal, and the order equals the matrix dimension. As before, chi(sigma_v)=chi(sigma_v^').


See also

Cyclic Group, Cyclic Group C2, Cyclic Group C4, Dihedral Group, Finite Group, Finite Group C2×C2×C2

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References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 244-245, 1985.Cotton, F. A. Chemical Applications of Group Theory, 3rd ed. New York: Wiley, 1990.

Cite this as:

Weisstein, Eric W. "Finite Group C_2×C_2." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FiniteGroupC2xC2.html

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