On SUBGROUPS of the ISOMETRY GROUP of the PLANE*

James Parks

SUNY Potsdam

Abstract This is a study of several subgroups of Iso, the group of all isometries of the

plane. These subgroups are generated by combinations of selected types of isometries. Almost all of the resulting subgroups so determined are semidirect products of known groups. Topological representations of these subgroups in a pair of open solid tori which represent the group Iso as an affine space are included for a more complete understanding of these subgroups and of Iso itself.

*See Errata to this article at http://www2.potsdam.edu/parksjm/Errata.htm

Mathematics Subject Classification 2000: 51A05, 14L17, 11E57, 54H11

Keywords and phrases: isometries, group of isometries of the plane, subgroups of isometries, semidirect products of groups, affine groups

1. Notation, Definitions, and Examples

This section includes a list of the four types of isometries of the plane, and several known examples of groups of isometries of the plane. We include them here for reference and to establish notation.

By the Classification Theorem for Isometries of the Plane: All isometries of the plane are compositions of at most three reflections in lines [3], it follows as a corollary that the four types of transformations listed below are the only isometries of the plane [3], [5].

(a) R_m denotes the reflection in line m. Reflections are opposite isometries (reverse orientations) which fix all points on line m. Reflections satisfy R_m^-1 = R_m.

(b) T_AB denotes the translation by the directed line segment AB. If A=B, then we define T_AA=i, the identity function. If A /= B, then T_AB = R_mR_n, where m//n, AB perpendicular to m, and the distance between lines m and n is |AB|/2. Translations are direct isometries (preserve orientations) and have no fixed points. Translations satisfy T_AB^-1= T_BA.

(c) R(P,q) denotes the rotation about the point P of angle q radians, 0 <= q < 2pi. Note that R(P,0) = i. Also, R(P,q) = R_mR_n, where m intersect n = {P} and the angle between m and n is q/2. Rotations are direct isometries and fix only the point P. Rotations satisfy R(P,q)^-1 = R(P,2pi-q). The rotation R_(P,pi), called a half-turn (or point-reflection), is denoted by R_P in these notes.

(d) G(m,AB) denotes the glide reflection about line m in the direction of AB, for AB//m, defined by G(m,AB) = R_mT_AB = T_ABR_m. Glide reflections are opposite isometries and if A/=B they have no fixed points. Glide reflections satisfy G(m,AB)^-1 = G(m,BA).

The following is a survey of examples of known groups of isometries of the plane.

(A) Iso denotes the group of all isometries of the plane under function composition. The composition of isometries is clearly an isometry, and the inverse of an isometry is an isometry, by the above list of types of isometries [3].

(B) Trans denotes the subgroup of all translations, <{T_AB}, ^. >, where T_CDT_AB = T_AB+CD, where the directed line segment AB+CD is taken in the usual vector sense.

Trans is clearly abelian, and it is also a normal subgroup of Iso, Trans<Iso. For let T = R_m, m arbitrary, then TT_ABT^-1 = T_CD where CD = R_m(AB), Figure 2. All other possible conjugations of T_AB will involve this product by the Classification Theorem for Isometries of the Plane given above [3].

This same argument holds more generally for any subgroup S of Iso, S<Iso, so by the observations above we have the following lemma which we will find useful below.

Lemma. A subgroup S of Iso is normal in Iso if TST^-1is in S for T an arbitrary reflection about a line m, T = R_m.

In addition, Trans is isomorphic to the group <R², + >, denoted by just R², Trans ~ R², by the isomorphism h:Trans -> R², given by h(T_AB) = (a, b), for (a, b) the translation coordinates of AB in a given coordinate system on R².

(C) Rot_P, denotes the subgroup of all rotations about the fixed point P, <{R(P,q)}, ^. >, which satisfies the composition rule R(P,q) R(P,r) = R(P,(q+r)mod2pi). This subgroup is isomorphic to the circle group S¹= <{(cosq, sinq): 0 <= q < 2pi}, ^. >, where ^. is the usual multiplication on the field of complex numbers, Rot_P ~ S¹. The isomorphism h:Rot_P -> S¹ is given by h(R(P,q) ) = (cosq, sinq).

Rot_P is clearly abelian, but it is not a normal subgroup in Iso, since we have T_PQR_(P,q)T_QP = R_(Q,q) is not in Rot_P, for R_(P,q) in Rot_P and P /= Q.

(D) Pres denotes the subgroup of orientation preserving (direct) isometries. It is a normal subgroup of Iso, Pres<Iso. For, if T in Pres, and S in Iso is orientation reversing, then S^-1 is also orientation reversing, so STS^-1 is orientation preserving and is thus in Pres (the case for S orientation preserving is obvious). Observe Pres is not abelian, since the half-turns satisfy RPRQ /= RQRP, for P/=Q, by the observation above on the rotation group Rot_P.

Pres has index 2 in Iso, the other coset being RPres, for R any orientation reversing (opposite) transformation, since if S is also orientation reversing, then S^-1R in Pres, so RPres = SPres. We will denote the coset RPres, for R orientation reversing, by Rev for orientation reversing isometries. Thus Iso = Pres U Rev.

2. Certain Subgroups of Iso

Several subgroups of Iso can be determined by taking combinations of selected isometries. As we will see, many of these subgroups are isomorphic to semidirect products of well-known groups, see also [1], [2], [5]. The examples below were motivated by Example 1 that follows, see also [3] pp.19-20. A generalization of this subgroup can be found in Example 4 below.

Example 1. Consider the subgroup J generated by all translations T_AB and all half-turns R_P for points P in the plane. This collection will form a group under composition, as the composition of two half-turns is a translation. This means that J is also generated by the set of all half-turns.

Observe that J is also generated by all translations and a single half-turn R_P, since any other half-turn R_Q can be generated by R_P by conjugation with translations T_PQR_PT_QP = R_Q.

Although the subgroup J is called the group generated by half-turns in [3], in view of the above observation on half-turns we will call J the group generated by all translations T_AB and a fixed half-turn R_P. The reason why we make this change will become apparent below.

J is nonabelian, since the composition of a translation and a half-turn is not commutative by the above observation on generating half-turns. J is normal in Iso, J<Iso, since R_mR_PR_m^-1 in J for arbitrary R_m by the Lemma in Section 1. If m is not on P, then R_mR_PR_m^-1= T_ABR_P, for A = R_P(X) and B = R_mR_PR_m^-1(X), for X /= P, Figure 1. Note that R = R_P in Figure 1.

Figure 1

Trans is a normal subgroup of J, since it is a subgroup of J and Trans<Iso by the above results. By Figure 2, conjugation by half-turns R_P is in fact inversion in Trans, R_PT_ABR_P^-1 = R_PT_ABR_P = T_AB^-1 = T_BA, for all T_AB. Note that R = R_P, and T = T_AB in Figure 2.

Figure 2

However, <{R_P, i}, ^. >, the 2-element subgroup generated by R_P, is not normal in J, since T_ABR_PT_BA /= R_P, for T_AB in Trans by the above observation on generating half-turns.

Also, J = HK for H = Trans and K = <{R_P, i}, ^. >, since products of the form R_PT_AB equal products of the form T_CDR_P for CD = BA, by the above observations. All other combinations are obvious.

The subgroups H and K also satisfy H intersect K = i.

Observe Trans has index 2 in J, since if R_P and R_Q are any two half-turns, then R_P^-1R_Q = R_PR_Q = T_+-2PQ, for +-2PQ a line segment parallel to PQ but possibly of opposite direction with |+-2PQ| = 2|PQ|, so R_PTrans = R_QTrans.

Thus J equals the semidirect product of the subgroups Trans and <{R_P, i}, ^. >, J = Trans ⋊ <{R_P, i}, ^. >, and by the isomorphisms above we have J ~ R² ⋊ Z₂. This group is the generalized dihedral group of R² and is denoted by Dih(R²) [5].

This proves the following result.

Theorem 1. The subgroup J, generated by all translations T_AB and a fixed half-turn R_P, is a normal subgroup of Iso, and is isomorphic to the generalized dihedral group of R², J ~ Dih(R²).

Example 2. Let m be a fixed line and AB a fixed directed line segment parallel to m, AB//m. Then H_m denotes the subgroup generated by the translation T_AB and the reflection R_m. This group contains the glide reflections G_(m,nAB) = T_nAB° R_m, nÎZ, for nAB the directed line segment parallel to and in the direction of AB with length |nAB| = n|AB| if n > 0, and in the opposite direction of AB with length |nAB| = -n|AB| if n < 0. It also contains the 2- element subgroup generated by R_m, <{ R_m, i}, ° >, and the infinite cyclic subgroup generated by powers of the translation T_AB, denoted <{T_nAB}_Z , ° >, since (T_AB)ⁿ = T_nAB, nÎZ. Call H_m the group generated by the defining transformations T_AB and R_m of a given glide reflection G_(m,AB).

Clearly H_m is abelian, thus the cyclic subgroup <{T_nAB}_Z , ° > and the 2-element subgroup <{ R_m, i}, ° > are normal in H_m. H_m is not normal in Iso, since the condition of the Lemma, Section 1, R_k°(T_nAB°R_m)°R_kÎH_m, is not satisfied for m not parallel to k. Also, observe that H_m = HK, for H = <{T_nAB}_Z , ° > and K = <{ R_m, i}, ° >, since it is abelian, and that HÇK = i. Thus we have H_m equal to the direct product of subgroups <{T_nAB}_Z ,° > and <{ R_m, i}, ° >, H_m = <{T_nAB}_Z , ° > ´ <{ R_m, i}, ° > » Z ´ Z₂, [1].

Similarly, let m be a fixed line in the plane, and let K_m denote the subgroup generated by the set of all translations T_AB of the plane and the reflection R_m. This group will contain all reflections R_n where m//n, since T_AB°R_m = R_n, for m//n, AB^m, and |AB| equal to twice the distance between m and n. Call K_m the group generated by all translations T_AB and the fixed reflection R_m.

If AB is not perpendicular to m, we get glide reflections in K_m of the form T_AB°R_n = G_(n,CD), where m//n and CD is the component of AB parallel to m. Since the product of reflections about parallel lines is a translation, R_m°R_n = T_CD, when m//n by the observation above, we have closure of the elements in K_m under composition. A similar argument holds for the glide reflections in K_m. K_m is not normal in Iso by an argument similar to that for H_m above.

The subgroup K_m is nonabelian, since T_AB°R_m ≠ R_m°T_AB, for AB not parallel to m, see Figure 3 (R = R_m, and T = T_AB). Clearly Trans is a normal subgroup of K_m, Trans<K_m, since Trans<Iso.

It follows that K_m = HK, where H = Trans and K = <{R_m, i}, ° >, since products of the form R_m°T_AB equal products of the form T_CD°R_m, for CD = R_m(AB). Also H and K satisfy HÇK = i. Thus K_m equals the semidirect product of the subgroups Trans and <{R_m, i},° >, which is isomorphic to the generalized dihedral group of R², K_m =Trans ⋊ <{R_m, i},°> » R² ⋊ Z₂ = Dih(R²).

Figure 3

Combining these results we have the following theorem.

Theorem 2. Subgroup H_m generated by the components T_AB and R_m of a given glide reflection G_(m,AB) is isomorphic to the cross product of Z and Z₂, H_m » Z ´ Z₂. The subgroup K_m generated by all translations T_AB and the fixed reflection R_m is isomorphic to the generalized dihedral group of R², K_m » R² ⋊ Z₂ = Dih(R²).

Example 3. Let P be a fixed point in the plane and consider the set of all reflections R_m where m is on P. This set generates a subgroup of Iso that we denote by G_P. The subgroup contains all rotations about P, since any rotation about P is a composition of two reflections about lines on P, R(P,q) = R_m° R_n, for m and n on P, properly chosen.

Recall R(P,0) = i and note that any element of the form R_m°R(P,q) can be written in the form R(P,ø)°R_m, since the equation R_m°R(P,q) = R(P,ø)°R_m has solution R(P,ø) = R_m°R(P,q)°R_m, for ø = 2π - q.

Therefore it is possible to fix a particular reflection R_l , for line l on P, and write every reflection about lines on P in terms of this reflection and a rotation about P, since if R_m is a reflection with m on P, then the equation R_m = R(P,ø)°R_l has solution R(P,ø) = R_m°R_l. Thus we call G_P the group generated by the set of all rotations about P, R(P,ø), and the fixed reflection R_l, line l on P.

The subgroup G_P is not normal in Iso, because R_n°R(P,q)°R_n is not in G_P when n is not on P.

However Rot_P is a normal subgroup of G_P, Rot_P<G_P, since we have

T°R(P,q)°T ^-1Î Rot_P, for all TÎG_P, by the observations on Rot_P above.

Also, the subgroup G_P is not abelian, since R_l°R(P,q)°R_l = R(P, 2π-q), Figure 4. Note that R = R(P,q) in Figure 4.

Figure 4.

Note the similarity of G_P with the dihedral group Dih_n of motions on a regular n-polyhedron P_n which contains the finite rotation subgroup Rot_P,n = <{R(P,2π/n) , … ,

R(P,2(n-1)π/n), i}, ° > as a normal subgroup. Since Rot_P,n < Rot_P, for all n, we see that G_P also contains an isomorphic copy of Dih_n, for all n. As noted above Rot_P<G_P, with conjugation by reflections equal to inversion in Rot_P.

In addition G_P = HK, for H = Rot_P and K = <{R_m, i}, ° >, by the above observations on products in G_P. Clearly Rot_P Ç <{R_m, i}, ° > = i. Therefore G_P is the semidirect product of subgroups Rot_P and <{R_m, i}, ° >, G_P = Rot_P ⋊ <{R_m, i}, ° > (cf. [1] Example 1, p.178).

From the examples above we thus know that G_P » S¹⋊ Z₂ = Dih(S¹), and we have the connection to dihedral groups alluded to above.

Thus we have proved the following result.

Theorem 3. The subgroup G_P generated by the set of all rotations about P, R(P,ø), and the fixed reflection R_l, line l on P, is isomorphic to the generalized dihedral group of S¹, G_P » Dih(S¹).

Example 4. Let P be a fixed point and n a positive integer, let L_P,n denote the subgroup of Iso generated by all translations T_AB and the rotation about P of order n R_(P,2π/n).

This subgroup contains all rotations of order n about any point Q in the plane, since R_(Q,2kπ/n) = T_PQ°R_(P,2kπ/n)°T_QP. As a consequence L_P,n is not abelian.

If n = 2 we have subgroup J in Example 1 above, L_P,2 = J. L_P,n clearly contains the subgroups Trans and Rot_P,n. The subgroup Trans is normal in L_P,n, Trans<L_P,n, since it is normal in Iso. However, Rot_P,n is not normal in L_P,n, by the above observation on rotations, thus L_P,n is not normal in Iso.

We also have Trans Ç Rot_P = i, and in addition L_P,n = HK, for H = Trans, K = Rot_P,n, by the normality of Trans in L_P,n. Thus we have L_P,n = Trans ⋊ Rot_P,n, and by the isomorphisms given above it follows that L_P,n » R² ⋊ Z_n.

Similarly, let P be a fixed point in the plane, and let L_P denote the subgroup generated by all translations T_AB and the rotations R_(P,q) about P, 0 ≤ q < 2π. As with L_P,n the subgroup L_P contains all rotations about any point Q in the plane, since R_(Q,q) = T_PQ°R_(P,q)°T_QP. Thus L_P is not abelian.

Trans is a normal subgroup in L_P, Trans<L_P, since Trans<Iso. Thus we have L_P = Trans ⋊ Rot_P, for Trans Ç Rot_P = i, Trans is normal in L_P, and Rot_P is a (non normal) subgroup of L_P. Also L_P = HK, for H = Trans, K = Rot_P, since for a given product R_(P,q)°T_AB, we have R_(P,q)°T_AB°R_(P,2π-q) = T_CD, where CD = R_(P,q)(AB), as above in L_P,n, so R_(P,q)°T_AB = T_CD°R_(P,q). Since Trans » R² and Rot_P » S¹, we then have L_P » R²⋊ S¹.

However, the alert reader is already aware that L_P = Pres, the normal subgroup of Iso of orientation preserving isometries, so we have the known result Pres » R²⋊ S¹ [5].

Thus we can also say that the group Iso is isomorphic to two copies of R² ⋊ S¹, that is Iso = Pres È Rev » R²⋊ S¹ È R²⋊ S¹. Since J<Iso, Example 1, we now have a pretty good idea what the quotient group Iso/J looks like. By the above results we have Iso/J = (Pres È Rev)/J » (R²⋊ S¹È R²⋊ S¹)/(R² ⋊ Z₂) = (R²⋊ S¹È R²⋊ S¹)/Dih(R²).

This means that the subgroup J is isomorphic to 2 copies of R², Example 1, one is the normal subgroup Trans » R² at q = 0 in S¹, and the other coset R_PR² at q = π in S¹, all in the subgroup Pres of Iso. So Iso/J is the set of cosets of J » R²⋊Z₂ of the form T°(R²⋊Z₂), where T is one of the isometries R(P,q) or R_m (note T_AB(R²⋊Z₂) = R²⋊Z₂).

For example, if T = R(P,q), then R(P,q)(R² ⋊ Z₂) is a pair of diametrically opposite copies of R² in the subgroup Pres at q in S¹, and if T = R_m, then R_m(R² ⋊ Z₂) is a pair of diametrically opposite copies of R² in the coset Rev. More on this in Section 3 below.

Notice that the copies R(P,q)(R² ⋊ Z₂) in Pres only use the range 0 ≤ q < π, since the copies of R² in J are diametrically opposite, so they will repeat when π ≤ q < 2π.

Also the the slope of the line m for R_m can be identified with the copies of R² in J by the position of J in Rev with respect to S¹, that is if m has slope q, then R_m(R²⋊Z₂) is in subgroup Rev at position q in S¹, 0 ≤ q < π. That the range of q is bounded by π follows from the observation that slopes q and q+π determine the same line m, and since the copies of R² in J are diametrically opposite, they will begin to repeat after they go through this range of q (see the topological representations below).

Combining these results we have the following theorem.

Theorem 4. The subgroup L_P,n, generated by all translations T_AB and the rotation about P of order n, R_(P,2π/n), is isomorphic to the semidirect product of Trans and the finite rotation subgroup Rot_P,n, L_P,n » R² ⋊ Z_n. The subgroup L_P, generated by all translations T_AB and all rotations about P, R_(P,q), for 0 ≤ q < 2π, is isomorphic to R²⋊ S¹, L_P » R²⋊ S¹» Pres. Hence we know that Iso is isomorphic to two copies of R² ⋊ S¹, Iso = Pres È Rev » R²⋊ S¹È R²⋊ S¹. Also, since J<Iso, it follows that Iso/J » (R²⋊ S¹È R²⋊ S¹)/(R² ⋊ Z₂) = (R²⋊ S¹ È R²⋊ S¹) / Dih(R²).

Example 5. By combining the subgroups J of Example 1 and K_m of Example 2 in a certain way we get a new subgroup, which we denote by J_m. This group is generated by three types of isometries: all translations of the plane T_AB, all half-turns R_P, and a reflection R_m about a fixed line m in the plane.

From the observations in Examples 1 and 2, the group J_m is also generated by all half-turns R_P and the single reflection R_m. As with the subgroup K_m the subgroup J_m contains all reflections R_n, for n//m, and all glide-reflections G_(n,AB), for n//m. Recall that the composition of any two half-turns R_P is a translation, and similarly, the composition of any two reflections R_n, n//m, is a translation.

We call J_m the subgroup generated by all half-turns R_P , for all P, and the reflection R_m, for m a fixed line. Compositions of reflections R_n, n//m, and half-turns R_P gives reflections R_l or products T_AB°R_l, where l^n, depending on where P is located.

Figure 5, Left shows the case for P on n, where R = R_P, and Figure 5, Right shows the case for P not on n. In the case of P not on n, Figure 5-Right, note that R_PR_n(DABC), is a translate of DR_l(A)R_l(B)R_l(C), the reflection of DABC about line l, by directed line segment R_l(A)R_PR_n(A).

Thus J_m also contains all reflections and glide-reflections about lines n, n^m. Since J is a nonabelian subgroup of J_m, from above, J_m is also nonabelian.

Figure 5

Observe J< J_m, since R_m^-1°R_nÎTrans, for all n//m, so R_mJ = R_nJ, and J has index 2 in J_m. Also Trans< J_m, since Trans is normal in Iso. Thus K_m< J_m, since if R_PÎ J_m, and R_nÎ K_m, then R_P°R_n°R_P = R_n’, for n//n’ and n’ = R_P(n), Figure 6 (where R = R_P).

However, J_m, is not normal in Iso, since R_n°R_m°R_n^-1 ≠ R_l, for l//m. Thus J_m = (NK)H, for N = Trans, K = <{R_P, i}, °>, and H = <{R_m, i}, ° >. But H = <{R_m, i}, ° > is not normal in J_m, by the observations above, so J_m is isomorphic to the semidirect product of J and Z₂, J_m » J ⋊ Z₂ » Dih(R²) ⋊ Z₂.

Figure 6

Combining these results we have the following theorem.

Theorem 5. The subgroup J_m generated by all half-turns R_P, all points P, and the reflection R_m, m a fixed line, is isomorphic to the semidirect product of J and Z₂, J_m » J ⋊ Z₂ » Dih(R²) ⋊ Z₂.

3. Topological Representations in Open Tori

It is instructive to compare the results in the examples above with topological results from affine geometry theory [4], [5].

By Example 4, we know that Pres » R²⋊ S¹ and Iso is algebraically equivalent to two copies of R²⋊ S¹. This is consistent with known results in affine space theory, where it is shown that Iso is topologically equivalent to the space R²´ O(2), O(2) the classical orthogonal group of degree 2. Also, since O(2) is the union of two copies of SO(2), the special orthogonal subgroup of degree 2, and SO(2) is in turn topologically equivalent to the circle S¹, we have O(2) topologically equivalent to 2 copies of S¹ [4], p.12.

Thus Iso is topologically equivalent to R²´ S¹È R²´ S¹, which is in turn topologically equivalent to the interior of two disjoint solid tori, intD²´ S¹È intD²´ S¹, since intD², the interior of the unit disc D², is topologically equivalent to R².

In this model one open solid torus represents the subgroup Pres, and the other torus represents the coset Rev, Figure 7, [4] pp. 19-21.

Pres Rev

Figure 7

The vertical open disc at x = +1 in the cut-away view of the Pres torus, Figure 8, represents the subgroup Trans. It’s complement in the torus Pres is the set of rotations. The open Mobius band intM inside the cut-away view of the coset torus Rev represents the set of all reflections, and it’s complement in the torus Rev represents the set of all glide reflections, and is topologically another open solid torus, ibid.

Pres Rev Figure 8

The subgroup J » R² ⋊ Z₂ in Example 1 is represented in this model by two vertical open circular discs (cross sections) on diametrically opposite sides of the open torus Pres. One disc at x = +1, for q = 0 in S¹ (to the right) represents the subgroup Trans » R², and the other disc at x = -1, for q = π in S¹ (to the left) represents the coset R_PTrans » R_PR². Figure 9 shows the cut-away view of subgroup J.

Pres

Figure 9

It follows that the quotient group Iso/J looks like copies of J indexed by q as above in both subgroups Pres and Rev over the range 0 ≤ q < π, as previously described in Example 4 above.

The subgroup K_m » R² ⋊ Z₂ in Example 2 is represented by two open discs. One at x = +1 in torus Pres, which represents the normal subgroup Trans » R², and the other coset in the torus Rev, a disc and a line, shown in a cut-away view on the right in Figure 10. The line in the torus Rev represents the reflections R_n, for n//m, and the complement of the line in this disc represents the glide reflections G_(n,AB), for n//m. The line in torus Rev is a cross-section of the open Mobius band intM determined by m, Figure 8.

Pres Rev

Figure 10

The subgroup G_P » S¹⋊ Z₂ of Example 3 corresponds to a pair of (horizontal) circles that are copies of S¹, one in each torus. The circle in torus Pres represents the rotation group Rot_P. The circle in torus Rev is in the Mobius band intM and represents the coset R_mRot_P, see the cut away views in Figure 11.

Pres Rev

Figure 11

The subgroup J_m » (R² ⋊ Z₂) ⋊ Z₂ = Dih(R²) ⋊ Z₂ » J ⋊ Z₂, Example 5, corresponds to a copy of J in the torus Pres, Figure 9, and a coset of J in the torus Rev, Figure 12. The line on the right side of the torus Rev represents the reflections R_n, for n//m. It is contained in the disc that contains the representative for line m. The line opposite to the left in torus Rev represents the reflections R_l, for l^m. The complement of these 2 lines in the discs shown in the torus Rev represent the glide reflections G_(n,AB), n//m, and G_(l,AB), l^m, respectively. These lines are cross-sections of the open Mobius band intM, Figure 8.

Pres Rev

Figure 12

References

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[2] S. MacLane & G. Birkhoff, Algebra, 4^th printing, MacMillan Co., 1970.

[3] G. Martin, Transformation Geometry, 4^th printing, S-V, 1982.

[4] E. Rees, Notes on Geometry, 2^nd printing, S-V, 1988.

[5] MathWorld: http://mathworld.wolfram.com