Page 86 - DMTH403_ABSTRACT_ALGEBRA

Unit 6: Group Isomorphism

If (G, *) is a locally finite group that is isomorphic to (H,  ), then (H,  ) is also locally Notes

finite.
We also go through that ‘group properties’ are always preserved by isomorphisms.

Cyclic Groups

All cyclic groups of a given order are isomorphic to ( , + ).
n
n
Let G be a cyclic group and n be the order of G. G is then the group generated by < x > = {e,x,...,
x n – 1 }. We will show that
G  ( , + )
n
n
Define
 : G   = {0, 1,..., n – 1}, so that (x ) = a. Clearly,  is bijective.
a
n
Then
(x . x ) = (x ) = a + b = (x ) + (x ) which proves that G   , +n.
a+b
a
b
b
a
n
n
Consequences
From the definition, it follows that any isomorphism f : G  H will map the identity element of
G to the identity element of H,
f(e ) = e
G H
that it will map inverses to inverses,
f(u ) = [f(u)] –1
–1
and more generally, nth powers to nth powers,
f(u ) = [f(u)] n
n
for all u in G, and that the inverse map f : H  G is also a group isomorphism.
–1
The relation “being isomorphic” satisfies all the axioms of an equivalence relation. If f is an
isomorphism between two groups G and H, then everything that is true about G that is only
related to the group structure can be translated via f into a true ditto statement about H, and vice
versa.

Self Assessment

1. Let H be a subgroup of a Group G. Then H  G, i(h) = h is a homomorphism. This function
is called the .................
(a) inclusion map (b) normal function
(c) cyclic (d) abelian
2. gof (x, y) is equal to:

(a) gof(x) . gof(y) (b) gof(x) + gof(y)
(c) gof(x ) . gof(y ) (d) gof(x) . gof(y )
-1
-1
-1
3. Let f : G  G be a group homomorphism thus her f is a ................. of G.
1
2
(a) subgroup (b) normal
(c) cyclic (d) abelian

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