6. ISOMORPHISMS
77
Theorem (3). ⇡ is an equivalence relation on the set G of all groups.
Proof.
(1) G ⇡ G. Define
: G ! G by (x) = x. This is clear.
(2) Suppose G ⇡ H. If : G ! H is the isomrphism and, for g 2 G,
(g) = h, then 1 : H ! G where 1(h) = g is 1–1 and onto.
Suppose a, b 2 H. Since is onto, 9 ↵,
Then (↵ ) = (↵) ( ) = ab, so
1
(ab) =
1
(↵) ( ) =
1
1
(↵) = a and ( ) = b.
(↵ ) = ↵ =
1
Thus
2G3
(↵)
1
( ) =
is an isomorphism and H ⇡ G.
(3) Suppose G ⇡ H and H ⇡ K with : G ! H and
isomorphisms. Then
: G ! K is 1–1 and onto.
For all a, b 2 G,
⇥
⇤
⇥
⇤
⇥
⇤ ⇥
⇤ ⇥
( )(ab) =
(ab) =
(a) (b) =
(a)
(b) = (
Thus G ⇡ K and ⇡ is an equivalence relation on G.
1
(a)
1
(b).
: H ! K the
⇤⇥
)(a) (
⇤
)(b) .
⇤
Note. Thus, when two groups are isomorphic, they are in some sense equal.
They di↵er in that their elements are named di↵erently. Knowing of a computation in one group, the isomorphism allows us to perform the analagous
computation in the other group.