Isomorphisms and Isomorphic Groups (10/9) We can now say what we mean by two groups being “the same” even though their operations and elements may look.

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Isomorphisms and Isomorphic Groups (10/9) We can now say what we mean by two groups being “the same” even though their operations and elements may look different. For example, we have been saying all along that D 3 and S 3 are “the same” even though their elements are labeled quite differently. The same is true, for example, of Z (under + of course) and the subgroup  2  of Q + (under times). What do they have in common? What is their basic structure? What about the subgroup {R 0, R 180, H, V} of D 4 and the subgroup {(1), (1 2), (4 5), (1 2)(4 5)} of S 5 ? What basic structure do they share?

Isomorphisms Definition. Let G and G be groups. A function  from G to G is call an isomorphism if:  is one-to-one and onto, and for all a and b in G,  (a b) =  (a)  (b). This latter condition says that  is “operation preserving”, or that it “respects the two operations”. Note that on the left the operation is occurring in G, but on the right it is occurring in G.

Examples of Isomorphisms Find an isomorphism from D 3 to S 3. Find an isomorphism from Z to 2Z. And so, in infinite groups, a group can be isomorphic of a proper subgroup of itself! Why is this not possible among finite groups? Find an isomorphism of Z to  2  in Q +. Let G =  a  be an abstract infinite cyclic group. Find an isomorphism from Z to G. Find an isomorphism of Z 4 to U(10). (Try to) find an isomorphism of Z 4 to U(8). If |G|  | G|, can an isomorphism between them exist?

Isomorphic Groups Definition. Two groups G and G are said to be isomorphic if there exists an isomorphism between them. So what examples do we already have of isomorphic groups? How do you show that two groups are isomorphic? Answer: Display an explicit isomorphism between them! How do you show that two groups are not isomorphic? Answer: Find a group-theoretic property that they do not share. For example: Order of groupsOrders of elements in the groups Cyclic versus non-cyclic Abelian versus non-abelian Numbers of subgroup of different orders Etc.

Examples of Non-Isomorphic Groups Why are the follow pairs of groups not isomorphic? Z 4 and Z 8 Z 4 and U(8) Z 8 and D 4 S 4 and Z 24 S 4 and D 12 In D 4,  R 90  and {R 0, R 180, H, V}. Z and Q + Q + and R + R and GL(2, R) Q + and Q*