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5.8. SOME FACTS FROM GROUP THEORY 225

b)a central cylinder if μ = 0, p < n; its canonical equation has the form

p
2

λ i x + c = 0;
i
i=1
c)a paraboloidal cylinder if μ ≠ 0, p < n – 1; in this case, the parallel translation along the
x n -axis by – c yields the canonical equation of a paraboloidal cylinder
2μ
p
2
λ i x + 2μx n = 0.
i
i=1

5.8. Some Facts from Group Theory
5.8.1. Groups and Their Basic Properties

5.8.1-1. Composition laws.

Let T be a mapping defined on ordered pairs a, b of elements of a set A and mapping each
pair a, b to an element c of A. In this case, one says that a composition law is definedonthe
set A. The element c A is called the composition of the elements a, b A and is denoted
by c = aTb.
A composition law is commonly expressed in one of the two forms:
1. Additive form: c = a + b; the corresponding composition law is called addition and c is
called the sum of a and b.
2. Multiplicative form: c = ab; the corresponding composition law is called multiplication
and c is called the product of a and b.
A composition law is said to be associative if

aT(bTc)= (aTb)Tc for all a, b, c A.

In additive form, this relation reads a +(b + c)=(a + b)+ c; and in multiplicative form,
a(bc)=(ab)c.
A composition law is said to be commutative if

aTb = bTa for all a, b A.

In additive form, this relation reads a + b = b + a; and in multiplicative form, ab = ba.
An element e of the set A is said to be neutral with respect to the composition law T if
aTe = a for any a A.
In the additive case, a neutral element is called a zero element, and in the multiplicative
case, an identity element.
An element b is called an inverse of a A if aTb = e. The inverse element is denoted
–1
by b = a .
In the additive case, the inverse element of a is called the negative of a and it is denoted
by –a.
Example 1. Addition and multiplication of real numbers are composition laws on the set of real numbers.
Both these laws are commutative. The neutral element for the addition is zero. The neutral element for the
multiplication is unity.