Page 85 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes Intuitively, group theorists view two isomorphic groups as follows: For every element g of a
group G, there exists an element h of H such that h ‘behaves in the same way’ as g (operates with
other elements of the group in the same way as g). For instance, if g generates G, then so does h.
This implies in particular that G and H are in bijective correspondence. So the definition of an
isomorphism is quite natural.
An isomorphism of groups may equivalently be defined as an invertible morphism in the
category of groups, where invertible here means has a two-sided inverse.

Examples:
1. The group of all real numbers with addition, (, +), is isomorphic to the group of all
positive real numbers with multiplication ( , ×):
+
(, +) (+, ×)
via the isomorphism
f(x) = e x
(see exponential function).

2. The group  of integers (with addition) is a subgroup of , and the factor group / is
isomorphic to the group S of complex numbers of absolute value 1 (with multiplication):
1
/ S 1

An isomorphism is given by
f(x + ) = e 2x 1
for every x in .
3. The Klein four-group is isomorphic to the direct product of two copies of  = /2
2
(see modular arithmetic), and can therefore be written  ×  . Another notation is Dih ,
2
2
2
because it is a dihedral group.
4. Generalizing this, for all odd n, Dih is isomorphic with the direct product of Dih and Z .
2
2n
n
5. If (G, *) is an infinite cyclic group, then (G, *) is isomorphic to the integers (with the
addition operation). From an algebraic point of view, this means that the set of all integers
(with the addition operation) is the ‘only’ infinite cyclic group.
Some groups can be proven to be isomorphic, relying on the axiom of choice, but the proof does
not indicate how to construct a concrete isomorphism.
1. The group (, +) is isomorphic to the group (, +) of all complex numbers with addition.
2. The group ( , ·) of non-zero complex numbers with multiplication as operation is
*
isomorphic to the group S mentioned above.
1
Properties

The kernel of an isomorphism from (G, *) to (H,  ), is always {e } where e is the identity
 G G
of the group (G, *)
If (G, *) is isomorphic to (H,  ), and if G is abelian then so is H.

If (G, *) is a group that is isomorphic to (H,  ) [where f is the isomorphism], then if a

belongs to G and has order n, then so does f(a).

78 LOVELY PROFESSIONAL UNIVERSITY