ABSTRACT ALGEBRAIC STRUCTURES OPERATIONS AND ...

58. Definition: Quotient induced by a homomorphism.
The equivalence relation mentioned in the preceding theorem is said to be induced
by f and we denote it by ∼f . The quotient structure induced by this
relation will be called the quotient structure induced by f and will be denoted
by A/f
4: Appendix: Tables.
4. 1: Symbols for sets.
In the table below X and Y are sets G, H are groups, M monoids, R rings,
L Fields and V, W linear spaces. If needed in the context we let X and Y be
equipped with topologies and we let V, W have inner products.
Notation Set
1 N natural numbers
2 Z integers
3 Q rational numbers
4 R real numbers
5 C complex numbers
6 M ∗ invertible elements in M
7 Zn classes of remainders Z/nZ
8 Un invertible elements in Zn, (Z ∗ n)
9 Lin(V, W ) linear mappings of V into W .
10 Iso(V, W ) isometries of V into W
11 Rot(2) rotations in the plane
12 Rot(3) rotations in space
13 Rfl(2) reflections in the plane
14 Rfl(3) reflections in space
15 T classes of remainders R/Z
16 Mat(m, n, L) m × n matrices over L
17 GL(n, L) invertible elements i Mat(n, n, L)
18 SL(n, L) {A ∈ GL(n, L)| det(A) = 1}
19 En unity matrix of dimension n
20 O(n) orthogonal matrices
21 SO(n) special orthogonal matrices
22 U(n) unitary matrices
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