Sn n-dimensional absolute space

En, Ln n-dimensional Euclidean and Lobachevsky's space

E neutral element

N set of natural numbers

R reflection

H subgroup

S n-fold rotation (n > 2)

{S1,S2,,Sn} set of generators

G×G1 direct product of groups G, G1

Cn, Dn cyclic and dihedral group

int interior

Cl closure

{p,q} regular tessellation

X, Y, Z translation

P, Q, O glide reflection

Gnst Bohm symbols of symmetry group categories

set of real numbers

k coefficient of similarity

K = K(k) dilatation with coefficient k

L = L(k,q) dilative rotation with coefficient k and dilative rotation angle q

M = M(k,m) dilative reflection with coefficient k and reflection line m

E2\{O} E2 plane without point O

RI inversion

ZI inversional reflection

SI inversional rotation

G/H, G/H/H1 group/subgroup symbols of antisymmetry and colored symmetry groups

[G:H]=N H is subgroup of index N of group G

n n-fold rotation

a translation

ã glide reflection

m reflection

g glide reflection

T two-fold rotation

(k,m) = 1 k, m are coprime numbers

p0 continuous line group of translations

[p,q] symmetry group of regular tessellation {p,q}

p00 continuous plane group of translations

mI inversion circle