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Suppose we have two groups G and H. H is not usually a subgroup of G but that’s
possible. Which of the following
properties of would you want G and H
to have if you were to consider them “essentially the same”? I.e. What does it mean if two groups are “isomorphic”? ·
The same number of elements in G as in H. ·
The name number of elements of order n in G as in
H. E.g. If G has 4 elements of order 3
then H has 4 elements of order 3. ·
The same subgroup structure. E.g. If G has an abelian subgroup G1
then H has a corresponding abelian subgroup H1 . In this case do you think that subgroup G1
and subgroup H1 should also have the same characteristics? ·
How about the multiplication? If a,b are elements of G and c,d are in H,
should we be able to say something about a·b
and c·d? All these considerations lead to the following formal
definition of isomorphism. We say G is
isomorphic to H if there is a 1-1,
onto map, say f, from G to H which has the following property: For any
a,b in G, f(a·b)=f(a)*f(b) Note that I’ve used two different multiplication symbols
because a·b is multiplication in G but f(a)*f(b) is multiplication in H. Since f:G→H then f(a) and f(b) are in H. Here are some consequences of this definition for you to
prove. For all of these a and b are in
G ·
If a is conjugate to b, then f(a)is conjugate to f(b). ·
If a and b commute, then f(a) and f(b) commute. ·
If e1 is the identity in G and e2
is the identity in H then f(e1)= e2. Take some care proving this. Hint: You may need to use “onto” ·
If o(a)=k (in
G), then o(f(a))=k in H . |